-- Mathlib/Data/Finset/Sigma.lean
theorem mem_sigmaLift (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (a : Sigma α) (b : Sigma β)
    (x : Sigma γ) :
    x ∈ sigmaLift f a b ↔ ∃ (ha : a.1 = x.1) (hb : b.1 = x.1), x.2 ∈ f (ha ▸ a.2) (hb ▸ b.2)

Proof state:
case mk.mk.inr.intro.refl.intro.refl
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
a : α x.1
b : β x.1
h : x.1 ≠ x.1
h✝ : x.snd ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) (⋯ ▸ ⟨x.1, b⟩.snd)
⊢ False

Tactic:
exact h rfl

Result:
no goals

Proof state:
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
a : α x.1
b : β x.1
hx : x.snd ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) (⋯ ▸ ⟨x.1, b⟩.snd)
⊢ (Embedding.sigmaMk ⟨x.1, b⟩.fst) x.snd = x

Tactic:
simp [Sigma.ext_iff]

Result:
no goals

Proof state:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
i : ι
a : α i
b : β i
x : γ i
hx : x ∈ f a b
⊢ ∃ (h : i = ⟨i, x⟩.fst) (h_1 : i = ⟨i, x⟩.fst), ⟨i, x⟩.snd ∈ f (⋯ ▸ a) (⋯ ▸ b)

Tactic:
exact ⟨rfl, rfl, hx⟩

Result:
no goals

Proof state:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ →
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨i, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨i, b⟩.snd)

Tactic:
simp_rw [sigmaLift]

Result:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ (x ∈ if h : True then map (Embedding.sigmaMk i) (f a b) else ∅) →
    ∃ (h : i = x.fst) (h_1 : i = x.fst), x.snd ∈ f (⋯ ▸ a) (⋯ ▸ b)

Proof state:
case mk.mk.inl.mpr
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ (∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨i, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨i, b⟩.snd)) →
    x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩

Tactic:
rintro ⟨⟨⟩, ⟨⟩, hx⟩

Result:
case mk.mk.inl.mpr.intro.refl.intro.refl
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
a : α x.1
b : β x.1
hx : x.snd ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) (⋯ ▸ ⟨x.1, b⟩.snd)
⊢ x ∈ sigmaLift f ⟨x.1, a⟩ ⟨x.1, b⟩

Proof state:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ (x ∈ if h : True then map (Embedding.sigmaMk i) (f a b) else ∅) →
    ∃ (h : i = x.fst) (h_1 : i = x.fst), x.snd ∈ f (⋯ ▸ a) (⋯ ▸ b)

Tactic:
simp only [dite_eq_ite, ite_true, mem_map, Embedding.sigmaMk_apply, forall_exists_index,
  and_imp]

Result:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ ∀ x_1 ∈ f a b, ⟨i, x_1⟩ = x → ∃ (h : i = x.fst) (h_1 : i = x.fst), x.snd ∈ f (⋯ ▸ a) (⋯ ▸ b)

Proof state:
case mk.mk.inr
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
j : ι
b : β j
h : i ≠ j
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨j, b⟩ ↔
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)

Tactic:
rw [sigmaLift, dif_neg h]

Result:
case mk.mk.inr
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
j : ι
b : β j
h : i ≠ j
⊢ x ∈ ∅ ↔ ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)

Proof state:
case mk.mk
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
j : ι
b : β j
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨j, b⟩ ↔
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)

Tactic:
obtain rfl | h := Decidable.eq_or_ne i j

Result:
case mk.mk.inl
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ ↔
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨i, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨i, b⟩.snd)

case mk.mk.inr
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
j : ι
b : β j
h : i ≠ j
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨j, b⟩ ↔
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)

Proof state:
case mk.mk.inl
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ ↔
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨i, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨i, b⟩.snd)

Tactic:
constructor

Result:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ →
    ∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨i, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨i, b⟩.snd)

case mk.mk.inl.mpr
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ (∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨i, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨i, b⟩.snd)) →
    x ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩

Proof state:
case mk.mk.inl.mpr.intro.refl.intro.refl
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
a : α x.1
b : β x.1
hx : x.snd ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) (⋯ ▸ ⟨x.1, b⟩.snd)
⊢ ∃ a_1 ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) ⟨x.1, b⟩.snd, (Embedding.sigmaMk ⟨x.1, b⟩.fst) a_1 = x

Tactic:
exact ⟨_, hx, by simp [Sigma.ext_iff]⟩

Result:
no goals

Proof state:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
b : β i
⊢ ∀ x_1 ∈ f a b, ⟨i, x_1⟩ = x → ∃ (h : i = x.fst) (h_1 : i = x.fst), x.snd ∈ f (⋯ ▸ a) (⋯ ▸ b)

Tactic:
rintro x hx rfl

Result:
case mk.mk.inl.mp
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
i : ι
a : α i
b : β i
x : γ i
hx : x ∈ f a b
⊢ ∃ (h : i = ⟨i, x⟩.fst) (h_1 : i = ⟨i, x⟩.fst), ⟨i, x⟩.snd ∈ f (⋯ ▸ a) (⋯ ▸ b)

Proof state:
case mk.mk.inr
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
i : ι
a : α i
j : ι
b : β j
h : i ≠ j
⊢ ¬∃ (ha : ⟨i, a⟩.fst = x.fst) (hb : ⟨j, b⟩.fst = x.fst), x.snd ∈ f (ha ▸ ⟨i, a⟩.snd) (hb ▸ ⟨j, b⟩.snd)

Tactic:
rintro ⟨⟨⟩, ⟨⟩, _⟩

Result:
case mk.mk.inr.intro.refl.intro.refl
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
γ : ι → Type u_4
inst✝ : DecidableEq ι
f : ⦃i : ι⦄ → α i → β i → Finset (γ i)
x : Sigma γ
a : α x.1
b : β x.1
h : x.1 ≠ x.1
h✝ : x.snd ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) (⋯ ▸ ⟨x.1, b⟩.snd)
⊢ False