-- Mathlib/MeasureTheory/PiSystem.lean
theorem mem_generatePiSystem_iUnion_elim {α β} {g : β → Set (Set α)} (h_pi : ∀ b, IsPiSystem (g b))
    (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b, g b)) :
    ∃ (T : Finset β) (f : β → Set α), (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b

Proof state:
case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_s b ∧ a ∈ f_t' b
⊢ a ∈ f_s b ∩ f_t' b

case neg
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∉ T_t'
h1 : a ∈ f_s b ∧ True
⊢ a ∈ f_s b ∧ True

case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : True ∧ a ∈ f_t' b
⊢ True ∧ a ∈ f_t' b

case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∈ T_t'
h1 : a ∈ f_s b ∩ f_t' b
⊢ a ∈ f_s b ∧ a ∈ f_t' b

case neg
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∈ T_s
hbt : b ∉ T_t'
h1 : a ∈ f_s b ∧ True
⊢ a ∈ f_s b ∧ True

case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
b : β
hbs : b ∉ T_s
hbt : b ∈ T_t'
h1 : True ∧ a ∈ f_t' b
⊢ True ∧ a ∈ f_t' b

Tactic:
all_goals exact h1

Result:
no goals

Proof state:
case base.intro.intro.intro
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t s : Set α
b : β
h_s_in_t' : s ∈ (fun b => g b) b
⊢ s = ⋂ b_1 ∈ {b}, (fun x => s) b_1 ∧ ∀ b_1 ∈ {b}, (fun x => s) b_1 ∈ g b_1

Tactic:
simpa using h_s_in_t'

Result:
no goals

Proof state:
case h.left
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
⊢ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b =
    ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅

Tactic:
ext a

Result:
case h.left.h
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
a : α
⊢ a ∈ (⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b ↔
    a ∈ ⋂ b ∈ T_s ∪ T_t', if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅

Proof state:
case h.right
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
⊢ (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b

Tactic:
split_ifs with hbs hbt hbt

Result:
case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∈ T_t'
⊢ f_s b ∩ f_t' b ∈ g b

case neg
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∈ T_s
hbt : b ∉ T_t'
⊢ f_s b ∈ g b

case pos
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∈ T_t'
⊢ f_t' b ∈ g b

case neg
α : Type u_3
β : Type u_4
g : β → Set (Set α)
h_pi : ∀ (b : β), IsPiSystem (g b)
t : Set α
T_t' : Finset β
f_t' : β → Set α
h_t' : ∀ b ∈ T_t', f_t' b ∈ g b
h_gen_t' : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_t', f_t' b)
T_s : Finset β
f_s : β → Set α
h_s : ∀ b ∈ T_s, f_s b ∈ g b
h_gen_s : generatePiSystem (⋃ b, g b) (⋂ b ∈ T_s, f_s b)
h_nonempty : ((⋂ b ∈ T_s, f_s b) ∩ ⋂ b ∈ T_t', f_t' b).Nonempty
b : β
h_b : b ∈ T_s ∪ T_t'
hbs : b ∉ T_s
hbt : b ∉ T_t'
⊢ ∅ ∈ g b