-- Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
theorem effectiveEpiFamily_tfae
    {α : Type} [Finite α] {B : CompHaus.{u}}
    (X : α → CompHaus.{u}) (π : (a : α) → (X a ⟶ B)) :
    TFAE
    [ EffectiveEpiFamily X π
    , Epi (Sigma.desc π)
    , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
    ]

Proof state:
case intro
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b) → Epi (Sigma.desc π)
x✝ : Epi (Sigma.desc π)
e : Function.Surjective ⇑(ConcreteCategory.hom (Sigma.desc π))
i : ∐ X ≅ finiteCoproduct X :=
  (colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X)
t : ↑(∐ X).toTop
q : (fun X => ↑X.toTop) (finiteCoproduct X) := (ConcreteCategory.hom i.hom) t
this : t = (ConcreteCategory.hom i.inv) ((ConcreteCategory.hom i.hom) t)
⊢ (ConcreteCategory.hom (π q.fst)) q.snd = (ConcreteCategory.hom (Sigma.desc π)) t

Tactic:
rw [this]

Result:
case intro
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b) → Epi (Sigma.desc π)
x✝ : Epi (Sigma.desc π)
e : Function.Surjective ⇑(ConcreteCategory.hom (Sigma.desc π))
i : ∐ X ≅ finiteCoproduct X :=
  (colimit.isColimit (Discrete.functor X)).coconePointUniqueUpToIso (finiteCoproduct.isColimit X)
t : ↑(∐ X).toTop
q : (fun X => ↑X.toTop) (finiteCoproduct X) := (ConcreteCategory.hom i.hom) t
this : t = (ConcreteCategory.hom i.inv) ((ConcreteCategory.hom i.hom) t)
⊢ (ConcreteCategory.hom (π q.fst)) q.snd =
    (ConcreteCategory.hom (Sigma.desc π)) ((ConcreteCategory.hom i.inv) ((ConcreteCategory.hom i.hom) t))

Proof state:
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
x✝ : ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b
e : ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b := x✝
⊢ Epi (Sigma.desc π)

Tactic:
rw [epi_iff_surjective]

Result:
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
x✝ : ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b
e : ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b := x✝
⊢ Function.Surjective ⇑(ConcreteCategory.hom (Sigma.desc π))

Proof state:
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
⊢ [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b].TFAE

Tactic:
tfae_have 3 → 2
| e => by
  rw [epi_iff_surjective]
  intro b
  obtain ⟨t, x, h⟩ := e b
  refine ⟨Sigma.ι X t x, ?_⟩
  change (Sigma.ι X t ≫ Sigma.desc π) x = _
  simpa using h

Result:
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b) → Epi (Sigma.desc π)
⊢ [EffectiveEpiFamily X π, Epi (Sigma.desc π), ∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b].TFAE