Theorem: ProbabilityTheory.iIndepFun.indepFun_finset
File: Mathlib/Probability/Independence/Basic.lean

tactic: let πSβ := Set.pi (Set.univ : Set S) ''
  Set.pi (Set.univ : Set S) fun i => { s : Set (β i) | MeasurableSet[m i] s }
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: let πS := { s : Set Ω | ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s }
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: have hπS_pi : IsPiSystem πS := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: have hπS_gen : (MeasurableSpace.pi.comap fun a (i : S) => f i a) = generateFrom πS := by
  rw [generateFrom_pi.symm, comap_generateFrom]
  congr
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: let πTβ := Set.pi (Set.univ : Set T) ''
    Set.pi (Set.univ : Set T) fun i => { s : Set (β i) | MeasurableSet[m i] s }
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: let πT := { s : Set Ω | ∃ t ∈ πTβ, (fun a (i : T) => f i a) ⁻¹' t = s }
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: have hπT_pi : IsPiSystem πT := by exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: have hπT_gen : (MeasurableSpace.pi.comap fun a (i : T) => f i a) = generateFrom πT := by
  rw [generateFrom_pi.symm, comap_generateFrom]
  congr
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: refine IndepSets.indep (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i))
  (Measurable.comap_le (measurable_pi_iff.mpr fun i => hf_meas i)) hπS_pi hπT_pi hπS_gen hπT_gen
  ?_
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
⊢ IndepFun (fun a i => f (↑i) a) fun a i => f (↑i) a

tactic: rintro _ _ ⟨s, ⟨sets_s, hs1, hs2⟩, rfl⟩ ⟨t, ⟨sets_t, ht1, ht2⟩, rfl⟩
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
⊢ IndepSets πS πT

tactic: simp only [Set.mem_univ_pi, Set.mem_setOf_eq] at hs1 ht1
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs1 : sets_s ∈ Set.pi Set.univ fun i => {s | MeasurableSet s}
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht1 : sets_t ∈ Set.pi Set.univ fun i => {s | MeasurableSet s}
ht2 : Set.pi Set.univ sets_t = t
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' s ∩ (fun a i => f (↑i) a) ⁻¹' t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' t)

tactic: rw [← hs2, ← ht2]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' s ∩ (fun a i => f (↑i) a) ⁻¹' t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' t)

tactic: exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
⊢ IsPiSystem πS

tactic: rw [generateFrom_pi.symm, comap_generateFrom]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
⊢ MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS

tactic: congr
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
⊢ generateFrom
      ((Set.preimage fun a i => f (↑i) a) '' (Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s})) =
    generateFrom πS

tactic: exact IsPiSystem.comap (@isPiSystem_pi _ _ ?_) _
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
⊢ IsPiSystem πT

tactic: rw [generateFrom_pi.symm, comap_generateFrom]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
⊢ MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT

tactic: congr
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
⊢ generateFrom
      ((Set.preimage fun a i => f (↑i) a) '' (Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s})) =
    generateFrom πT

tactic: let sets_s' : ∀ i : ι, Set (β i) := fun i =>
  dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: have h_sets_s'_eq : ∀ {i} (hi : i ∈ S), sets_s' i = sets_s ⟨i, hi⟩ := by
  intro i hi; simp_rw [dif_pos hi]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: have h_sets_s'_univ : ∀ {i} (_hi : i ∈ T), sets_s' i = Set.univ := by
  intro i hi; simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: let sets_t' : ∀ i : ι, Set (β i) := fun i =>
  dite (i ∈ T) (fun hi => sets_t ⟨i, hi⟩) fun _ => Set.univ
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: have h_sets_t'_univ : ∀ {i} (_hi : i ∈ S), sets_t' i = Set.univ := by
  intro i hi; simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: have h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i) := by
  intro i hi; rw [h_sets_s'_eq hi]; exact hs1 _
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: have h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i) := by
  intro i hi; simp_rw [dif_pos hi]; exact ht1 _
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf_Indep
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: rw [h_eq_inter_S, h_eq_inter_T, hf_Indep S h_meas_s', hf_Indep T h_meas_t']
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
⊢ ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s ∩ (fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t) =
    ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_s) * ↑↑μ ((fun a i => f (↑i) a) ⁻¹' Set.pi Set.univ sets_t)

tactic: rw [h_Inter_inter, hf_Indep (S ∪ T)]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ↑↑μ ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    (∏ i in S, ↑↑μ (f i ⁻¹' sets_s' i)) * ∏ i in T, ↑↑μ (f i ⁻¹' sets_t' i)

tactic: swap
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ∏ i in S ∪ T, ↑↑μ (f i ⁻¹' (sets_s' i ∩ sets_t' i)) =
    (∏ i in S, ↑↑μ (f i ⁻¹' sets_s' i)) * ∏ i in T, ↑↑μ (f i ⁻¹' sets_t' i)

case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i)

tactic: rw [Finset.prod_union hST]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ∏ i in S ∪ T, ↑↑μ (f i ⁻¹' (sets_s' i ∩ sets_t' i)) =
    (∏ i in S, ↑↑μ (f i ⁻¹' sets_s' i)) * ∏ i in T, ↑↑μ (f i ⁻¹' sets_t' i)

tactic: congr 1
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ (∏ x in S, ↑↑μ (f x ⁻¹' (sets_s' x ∩ sets_t' x))) * ∏ x in T, ↑↑μ (f x ⁻¹' (sets_s' x ∩ sets_t' x)) =
    (∏ i in S, ↑↑μ (f i ⁻¹' sets_s' i)) * ∏ i in T, ↑↑μ (f i ⁻¹' sets_t' i)

tactic: intro i hi
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }

tactic: simp_rw [dif_pos hi]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
i : ι
hi : i ∈ S
⊢ sets_s' i = sets_s { val := i, property := hi }

tactic: intro i hi
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
⊢ ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ

tactic: simp_rw [dif_neg (Finset.disjoint_right.mp hST hi)]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
i : ι
hi : i ∈ T
⊢ sets_s' i = Set.univ

tactic: intro i hi
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ

tactic: simp_rw [dif_neg (Finset.disjoint_left.mp hST hi)]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
i : ι
hi : i ∈ S
⊢ sets_t' i = Set.univ

tactic: intro i hi
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
⊢ ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)

tactic: rw [h_sets_s'_eq hi]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
i : ι
hi : i ∈ S
⊢ MeasurableSet (sets_s' i)

tactic: exact hs1 _
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
i : ι
hi : i ∈ S
⊢ MeasurableSet (sets_s { val := i, property := hi })

tactic: intro i hi
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
⊢ ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)

tactic: simp_rw [dif_pos hi]
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
i : ι
hi : i ∈ T
⊢ MeasurableSet (sets_t' i)

tactic: exact ht1 _
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
i : ι
hi : i ∈ T
⊢ MeasurableSet (sets_t { val := i, property := hi })

tactic: ext1 x
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
⊢ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i

tactic: simp_rw [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
state_before: case h
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
⊢ x ∈ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s ↔ x ∈ ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i

tactic: constructor <;> intro h
state_before: case h
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
⊢ (∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i) ↔
    ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ

tactic: intro i hi
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i
⊢ ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ

tactic: simp only [h_sets_s'_eq hi, Set.mem_preimage]
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i
i : ι
hi : i ∈ S
⊢ x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ

tactic: exact h ⟨i, hi⟩
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
h : ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i
i : ι
hi : i ∈ S
⊢ f i x ∈ sets_s { val := i, property := hi }

tactic: rintro ⟨i, hi⟩
state_before: case h.mpr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
h : ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ ∀ (i : { x // x ∈ S }), f (↑i) x ∈ sets_s i

tactic: specialize h i hi
state_before: case h.mpr.mk
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
h : ∀ (i : ι), i ∈ S → x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
i : ι
hi : i ∈ S
⊢ f (↑{ val := i, property := hi }) x ∈ sets_s { val := i, property := hi }

tactic: rwa [dif_pos hi] at h
state_before: case h.mpr.mk
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
x : Ω
i : ι
hi : i ∈ S
h : x ∈ f i ⁻¹' if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ f (↑{ val := i, property := hi }) x ∈ sets_s { val := i, property := hi }

tactic: ext1 x
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
⊢ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i

tactic: simp only [Set.mem_preimage, Set.mem_univ_pi, Set.mem_iInter]
state_before: case h
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
⊢ x ∈ (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t ↔ x ∈ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i

tactic: constructor <;> intro h
state_before: case h
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
⊢ (∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i) ↔
    ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: intro i hi
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i
⊢ ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: simp_rw [dif_pos hi]
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i
i : ι
hi : i ∈ T
⊢ f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: exact h ⟨i, hi⟩
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
h : ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i
i : ι
hi : i ∈ T
⊢ f i x ∈ sets_t { val := i, property := hi }

tactic: rintro ⟨i, hi⟩
state_before: case h.mpr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
h : ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ ∀ (i : { x // x ∈ T }), f (↑i) x ∈ sets_t i

tactic: specialize h i hi
state_before: case h.mpr.mk
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
h : ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
i : ι
hi : i ∈ T
⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi }

tactic: simp_rw [dif_pos hi] at h
state_before: case h.mpr.mk
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
i : ι
hi : i ∈ T
h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi }

tactic: exact h
state_before: case h.mpr.mk
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
x : Ω
i : ι
hi : i ∈ T
h : f i x ∈ sets_t { val := i, property := hi }
⊢ f (↑{ val := i, property := hi }) x ∈ sets_t { val := i, property := hi }

tactic: ext1 x
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
⊢ ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)

tactic: simp_rw [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
state_before: case h
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
⊢ (x ∈ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) ↔
    x ∈ ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)

tactic: constructor <;> intro h
state_before: case h
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
⊢ ((∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧
      ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ) ↔
    ∀ (i : ι),
      i ∈ S ∨ i ∈ T →
        f i x ∈
          (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
            if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: intro i hi
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
h :
  (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧
    ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ ∀ (i : ι),
    i ∈ S ∨ i ∈ T →
      f i x ∈
        (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
          if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: cases' hi with hiS hiT
state_before: case h.mp
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
h :
  (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧
    ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
i : ι
hi : i ∈ S ∨ i ∈ T
⊢ f i x ∈
    (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
      if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: replace h := h.1 i hiS
state_before: case h.mp.inl
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
h :
  (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧
    ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
i : ι
hiS : i ∈ S
⊢ f i x ∈
    (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
      if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: simp_rw [dif_pos hiS, dif_neg (Finset.disjoint_left.mp hST hiS)]
state_before: case h.mp.inl
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
i : ι
hiS : i ∈ S
h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ f i x ∈
    (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
      if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: exact ⟨by rwa [dif_pos hiS] at h, Set.mem_univ _⟩
state_before: case h.mp.inl
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
i : ι
hiS : i ∈ S
h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ f i x ∈ sets_s { val := i, property := hiS } ∩ Set.univ

tactic: rwa [dif_pos hiS] at h
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
i : ι
hiS : i ∈ S
h : f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
⊢ f i x ∈ sets_s { val := i, property := hiS }

tactic: replace h := h.2 i hiT
state_before: case h.mp.inr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
h :
  (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧
    ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
i : ι
hiT : i ∈ T
⊢ f i x ∈
    (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
      if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: simp_rw [dif_pos hiT, dif_neg (Finset.disjoint_right.mp hST hiT)]
state_before: case h.mp.inr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
i : ι
hiT : i ∈ T
h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ f i x ∈
    (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
      if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: exact ⟨Set.mem_univ _, by rwa [dif_pos hiT] at h⟩
state_before: case h.mp.inr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
i : ι
hiT : i ∈ T
h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ f i x ∈ Set.univ ∩ sets_t { val := i, property := hiT }

tactic: rwa [dif_pos hiT] at h
state_before: Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
i : ι
hiT : i ∈ T
h : f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ f i x ∈ sets_t { val := i, property := hiT }

tactic: exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩
state_before: case h.mpr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
x : Ω
h :
  ∀ (i : ι),
    i ∈ S ∨ i ∈ T →
      f i x ∈
        (if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∩
          if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
⊢ (∀ (i : ι), i ∈ S → f i x ∈ if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ) ∧
    ∀ (i : ι), i ∈ T → f i x ∈ if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ

tactic: intro i hi_mem
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ∀ (i : ι), i ∈ S ∪ T → MeasurableSet (sets_s' i ∩ sets_t' i)

tactic: rw [Finset.mem_union] at hi_mem
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi_mem : i ∈ S ∪ T
⊢ MeasurableSet (sets_s' i ∩ sets_t' i)

tactic: cases' hi_mem with hi_mem hi_mem
state_before: case intro.intro.intro.intro.intro.intro.intro.intro
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi_mem : i ∈ S ∨ i ∈ T
⊢ MeasurableSet (sets_s' i ∩ sets_t' i)

tactic: rw [h_sets_t'_univ hi_mem, Set.inter_univ]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.inl
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (sets_s' i ∩ sets_t' i)

tactic: exact h_meas_s' i hi_mem
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.inl
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi_mem : i ∈ S
⊢ MeasurableSet (sets_s' i)

tactic: rw [h_sets_s'_univ hi_mem, Set.univ_inter]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.inr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi_mem : i ∈ T
⊢ MeasurableSet (sets_s' i ∩ sets_t' i)

tactic: exact h_meas_t' i hi_mem
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.inr
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi_mem : i ∈ T
⊢ MeasurableSet (sets_t' i)

tactic: refine' Finset.prod_congr rfl fun i hi => _
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.e_a
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ∏ x in S, ↑↑μ (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i in S, ↑↑μ (f i ⁻¹' sets_s' i)

tactic: rw [h_sets_t'_univ hi, Set.inter_univ]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.e_a
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi : i ∈ S
⊢ ↑↑μ (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ↑↑μ (f i ⁻¹' sets_s' i)

tactic: refine' Finset.prod_congr rfl fun i hi => _
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.e_a
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
⊢ ∏ x in T, ↑↑μ (f x ⁻¹' (sets_s' x ∩ sets_t' x)) = ∏ i in T, ↑↑μ (f i ⁻¹' sets_t' i)

tactic: rw [h_sets_s'_univ hi, Set.univ_inter]
state_before: case intro.intro.intro.intro.intro.intro.intro.intro.e_a
Ω : Type u_1
ι✝ : Type ?u.3366215
β✝ : Type ?u.3366218
β' : Type ?u.3366221
γ : Type ?u.3366224
γ' : Type ?u.3366227
mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
f✝ : Ω → β✝
g : Ω → β'
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
β : ι → Type u_3
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ (i : ι), i ∈ S → MeasurableSet (sets i)) →
      ↑↑μ (⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets i) = ∏ i in S, ↑↑μ (f i ⁻¹' sets i)
hf_meas : ∀ (i : ι), Measurable (f i)
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t, t ∈ πSβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := Set.pi Set.univ '' Set.pi Set.univ fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t, t ∈ πTβ ∧ (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : Set.pi Set.univ sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : Set.pi Set.univ sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s { val := i, property := hi } else Set.univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s { val := i, property := hi }
h_sets_s'_univ : ∀ {i : ι}, i ∈ T → sets_s' i = Set.univ
sets_t' : (i : ι) → Set (β i) := fun i => if hi : i ∈ T then sets_t { val := i, property := hi } else Set.univ
h_sets_t'_univ : ∀ {i : ι}, i ∈ S → sets_t' i = Set.univ
h_meas_s' : ∀ (i : ι), i ∈ S → MeasurableSet (sets_s' i)
h_meas_t' : ∀ (i : ι), i ∈ T → MeasurableSet (sets_t' i)
h_eq_inter_S : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_s = ⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i
h_eq_inter_T : (fun ω i => f (↑i) ω) ⁻¹' Set.pi Set.univ sets_t = ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i
h_Inter_inter :
  ((⋂ (i : ι) (_ : i ∈ S), f i ⁻¹' sets_s' i) ∩ ⋂ (i : ι) (_ : i ∈ T), f i ⁻¹' sets_t' i) =
    ⋂ (i : ι) (_ : i ∈ S ∪ T), f i ⁻¹' (sets_s' i ∩ sets_t' i)
i : ι
hi : i ∈ T
⊢ ↑↑μ (f i ⁻¹' (sets_s' i ∩ sets_t' i)) = ↑↑μ (f i ⁻¹' sets_t' i)