-- Mathlib/MeasureTheory/PiSystem.lean
theorem mem_generatePiSystem_iUnion_elim' {α β} {g : β → Set (Set α)} {s : Set β}
    (h_pi : ∀ b ∈ s, IsPiSystem (g b)) (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)) :
    ∃ (T : Finset β) (f : β → Set α), ↑T ⊆ s ∧ (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b

Proof state:
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
⊢ ↑(Finset.image (fun x => ↑x) T) ⊆ s

Tactic:
simp

Result:
no goals

Proof state:
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
t : Set α
h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)
⊢ ⋃ b, (g ∘ Subtype.val) b = ⋃ b ∈ s, g b

Tactic:
ext x

Result:
case h
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
t : Set α
h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)
x : Set α
⊢ x ∈ ⋃ b, (g ∘ Subtype.val) b ↔ x ∈ ⋃ b ∈ s, g b

Proof state:
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
⊢ (∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨a_1, b⟩) →
    ∀ (a_2 : β) (b : s a_2), ⟨a_2, b⟩ ∈ T → a ∈ f ⟨a_2, b⟩

Tactic:
intro h1 b h_b h_b_in_T

Result:
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨a_1, b⟩
b : β
h_b : s b
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ f ⟨b, h_b⟩

Proof state:
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨a_1, b⟩
b : β
h_b : s b
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ f ⟨b, h_b⟩ → a ∈ f ⟨b, h_b⟩

Tactic:
apply id

Result:
no goals

Proof state:
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
t : Set α
h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)
this : t ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
⊢ ∃ T f, ↑T ⊆ s ∧ t = ⋂ b ∈ T, f b ∧ ∀ b ∈ T, f b ∈ g b

Tactic:
rcases @mem_generatePiSystem_iUnion_elim α (Subtype s) (g ∘ Subtype.val)
    (fun b => h_pi b.val b.property) t this with
  ⟨T, ⟨f, ⟨rfl, h_t'⟩⟩⟩

Result:
case intro.intro.intro
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
⊢ ∃ T_1 f_1, ↑T_1 ⊆ s ∧ ⋂ b ∈ T, f b = ⋂ b ∈ T_1, f_1 b ∧ ∀ b ∈ T_1, f_1 b ∈ g b

Proof state:
case intro.intro.intro.refine_2.intro
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b_w : b ∈ s
h_b_h : ⟨b, ⋯⟩ ∈ T
⊢ Function.extend (fun x => ↑x) f (fun x => ∅) b ∈ g b

Tactic:
have h_b_alt : b = (Subtype.mk b h_b_w).val := rfl

Result:
case intro.intro.intro.refine_2.intro
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b_w : b ∈ s
h_b_h : ⟨b, ⋯⟩ ∈ T
h_b_alt : b = ↑⟨b, h_b_w⟩
⊢ Function.extend (fun x => ↑x) f (fun x => ∅) b ∈ g b

Proof state:
case intro.intro.intro.refine_2
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
⊢ ∀ b ∈ Finset.image (fun x => ↑x) T, Function.extend (fun x => ↑x) f (fun x => ∅) b ∈ g b

Tactic:
intros b h_b

Result:
case intro.intro.intro.refine_2
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b : b ∈ Finset.image (fun x => ↑x) T
⊢ Function.extend (fun x => ↑x) f (fun x => ∅) b ∈ g b

Proof state:
case intro.intro.intro.refine_2.intro
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b_w : b ∈ s
h_b_h : ⟨b, ⋯⟩ ∈ T
h_b_alt : b = ↑⟨b, h_b_w⟩
⊢ Function.extend (fun x => ↑x) f (fun x => ∅) b ∈ g b

Tactic:
rw [h_b_alt, Subtype.val_injective.extend_apply]

Result:
case intro.intro.intro.refine_2.intro
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
b : β
h_b_w : b ∈ s
h_b_h : ⟨b, ⋯⟩ ∈ T
h_b_alt : b = ↑⟨b, h_b_w⟩
⊢ f ⟨b, h_b_w⟩ ∈ g ↑⟨b, h_b_w⟩

Proof state:
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨a_1, b⟩
b : β
h_b : s b
h_b_in_T : ⟨b, h_b⟩ ∈ T
⊢ a ∈ f ⟨b, h_b⟩

Tactic:
have h2 := h1 b h_b h_b_in_T

Result:
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
h1 : ∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨a_1, b⟩
b : β
h_b : s b
h_b_in_T : ⟨b, h_b⟩ ∈ T
h2 : a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨b, h_b⟩
⊢ a ∈ f ⟨b, h_b⟩