-- Mathlib/Topology/Category/LightProfinite/Basic.lean
theorem epi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
    Epi f ↔ Function.Surjective f

Proof state:
case mp.intro.intro.intro.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
H : h = g
⊢ False

Tactic:
apply_fun fun e => (e y).down at H

Result:
case mp.intro.intro.intro.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
H : ((ConcreteCategory.hom h) y).down = ((ConcreteCategory.hom g) y).down
⊢ False

Proof state:
case mp.intro.intro.intro.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
H : ((ConcreteCategory.hom h).toFun y).down = { down := 0 }.down
⊢ False

Tactic:
exact top_ne_bot H

Result:
no goals

Proof state:
case mp.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
⊢ False

Tactic:
have hyU : y ∈ U := by
  refine Set.mem_compl ?_
  rintro ⟨y', hy'⟩
  exact hy y' hy'

Result:
case mp.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
⊢ False

Proof state:
case mp
X Y : LightProfinite
f : X ⟶ Y
⊢ (∃ b, ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ b) → ¬Epi f

Tactic:
rintro ⟨y, hy⟩ hf

Result:
case mp.intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
⊢ False

Proof state:
case h.h.h.h.hnc
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
x : ↑X.toTop
⊢ (ConcreteCategory.hom f).toFun x ∉ V

Tactic:
refine mt (fun α => hVU α) ?_

Result:
case h.h.h.h.hnc
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
x : ↑X.toTop
⊢ (ConcreteCategory.hom f).toFun x ∉ U

Proof state:
case intro
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
y' : (fun X => ↑X.toTop) X
hy' : (ConcreteCategory.hom f) y' = y
⊢ False

Tactic:
exact hy y' hy'

Result:
no goals

Proof state:
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
⊢ h = g

Tactic:
rw [← cancel_epi f]

Result:
X Y : LightProfinite
f : X ⟶ Y
y : ↑Y.toTop
hy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y
hf : Epi f
C : Set ((fun X => ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)
hC : IsClosed C
U : Set ((fun X => ↑X.toTop) Y) := Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set ↑Y.toTop
hV : V ∈ {s | IsClopen s}
hyV : y ∈ V
hVU : V ⊆ U
Z : LightProfinite := of (ULift.{u, 0} (Fin 2))
g : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := ⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen hV)), continuous_toFun := ⋯ }
h : Y ⟶ Z :=
  ofHom (fun X => TotallyDisconnectedSpace ↑X ∧ SecondCountableTopology ↑X)
    { toFun := fun x => { down := 1 }, continuous_toFun := ⋯ }
⊢ f ≫ h = f ≫ g