-- Module: Mathlib.Probability.Independence.Kernel.IndepFun
Goal:
α : Type u_1
Ω : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : Filter.EventuallyEq.{u_1, u_2} (β := Measure Ω) (ae μ) (⇑κ : (a : α) → Measure Ω) (⇑η : (a : α) → Measure Ω)
hη : IsMarkovKernel η
πSβ : Set (Set ((i : ↥S) → β (↑i : ι))) :=
  Set.pi (α := fun (i : ↥S) ↦ β (↑i : ι)) univ ''
    Set.pi (α := fun (i : ↥S) ↦ Set (β (↑i : ι))) univ fun (i : ↥S) ↦ {s : Set (β (↑i : ι)) | MeasurableSet s}
πS : Set (Set Ω) := {s : Set Ω | ∃ t ∈ πSβ, (fun (a : Ω) (i : ↥S) ↦ f (↑i : ι) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen :
  MeasurableSpace.comap (β := (i : ↥S) → β (↑i : ι)) (fun (a : Ω) (i : ↥S) ↦ f (↑i : ι) a)
      (MeasurableSpace.pi (X := fun (a : ↥S) ↦ β (↑a : ι))) =
    generateFrom πS
πTβ : Set (Set ((i : ↥T) → β (↑i : ι))) :=
  Set.pi (α := fun (i : ↥T) ↦ β (↑i : ι)) univ ''
    Set.pi (α := fun (i : ↥T) ↦ Set (β (↑i : ι))) univ fun (i : ↥T) ↦ {s : Set (β (↑i : ι)) | MeasurableSet s}
πT : Set (Set Ω) := {s : Set Ω | ∃ t ∈ πTβ, (fun (a : Ω) (i : ↥T) ↦ f (↑i : ι) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen :
  MeasurableSpace.comap (β := (i : ↥T) → β (↑i : ι)) (fun (a : Ω) (i : ↥T) ↦ f (↑i : ι) a)
      (MeasurableSpace.pi (X := fun (a : ↥T) ↦ β (↑a : ι))) =
    generateFrom πT
s : Set ((i : ↥S) → β (↑i : ι))
sets_s : (i : ↥S) → Set (β (↑i : ι))
hs2 : Set.pi (α := fun (i : ↥S) ↦ β (↑i : ι)) univ sets_s = s
t : Set ((i : ↥T) → β (↑i : ι))
sets_t : (i : ↥T) → Set (β (↑i : ι))
ht2 : Set.pi (α := fun (i : ↥T) ↦ β (↑i : ι)) univ sets_t = t
hs1 : ∀ (i : ↥S), MeasurableSet (sets_s i)
ht1 : ∀ (i : ↥T), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) :=
  fun (i : ι) ↦ if hi : i ∈ S then sets_s (Subtype.mk (p := fun (x : ι) ↦ x ∈ S) i hi) else univ
h_sets_s'_eq : ∀ {i : ι} (hi : i ∈ S), sets_s' i = sets_s (Subtype.mk (p := fun (x : ι) ↦ x ∈ S) i hi)
h_sets_s'_univ : ∀ {i : ι} (_hi : i ∈ T), sets_s' i = univ
sets_t' : (i : ι) → Set (β i) :=
  fun (i : ι) ↦ if hi : i ∈ T then sets_t (Subtype.mk (p := fun (x : ι) ↦ x ∈ T) i hi) else univ
h_sets_t'_univ : ∀ {i : ι} (_hi : i ∈ S), sets_t' i = univ
h_meas_s' : ∀ i ∈ S, MeasurableSet (sets_s' i)
h_meas_t' : ∀ i ∈ T, MeasurableSet (sets_t' i)
h_eq_inter_S :
  Eq.{u_2 + 1} (α := Set Ω)
    ((fun (ω : Ω) (i : ↥S) ↦ f (↑i : ι) ω) ⁻¹' Set.pi (α := fun (i : ↥S) ↦ β (↑i : ι)) univ sets_s)
    (⋂ i ∈ S, f i ⁻¹' sets_s' i)
h_eq_inter_T :
  Eq.{u_2 + 1} (α := Set Ω)
    ((fun (ω : Ω) (i : ↥T) ↦ f (↑i : ι) ω) ⁻¹' Set.pi (α := fun (i : ↥T) ↦ β (↑i : ι)) univ sets_t)
    (⋂ i ∈ T, f i ⁻¹' sets_t' i)
hf_Indep :
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)} (_H : ∀ i ∈ S, MeasurableSet (sets i)),
    ∀ᵐ (a : α) ∂μ,
      Eq (α := ENNReal)
        ((((η : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ENNReal) (⋂ i ∈ S, f i ⁻¹' sets i) : ENNReal)
        (∏ i ∈ S, (((η : (a : α) → Measure Ω) a : Measure Ω) : (a : Set Ω) → ENNReal) (f i ⁻¹' sets i) : ENNReal)
⊢ Eq.{u_2 + 1} (α := Set Ω) ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i)
    (⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i))

Tactic:
have h_Inter_inter :
    ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
      ⋂ i ∈ S ∪ T, f i ⁻¹' (sets_s' i ∩ sets_t' i) := by
    ext1 x
    simp_rw [Set.mem_inter_iff, Set.mem_iInter, Set.mem_preimage, Finset.mem_union]
    constructor <;> intro h
    · grind
    · exact ⟨fun i hi => (h i (Or.inl hi)).1, fun i hi => (h i (Or.inr hi)).2⟩