-- Module: Mathlib.Topology.Category.Profinite.Basic
Goal:
X Y : Profinite
f : X ⟶ Y
y : (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)
hy :
  ∀ (a : (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)),
    (ConcreteCategory.hom (FC :=
            fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
            C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
              (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
            (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
            (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) f :
          (a : (↑X.toTop : Type u)) → (↑Y.toTop : Type u))
        a ≠
      y
hf : Epi f
C : Set
  ((fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
      (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
    Y) :=
  Set.range (ι :=
    (fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
      X)
    (⇑(ConcreteCategory.hom (FC :=
          fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
          C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
            (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
          (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) f) :
      (a : (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u)) X) →
        (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u)) Y)
hC : IsClosed C
U : Set
  ((fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
      (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
    Y) :=
  Cᶜ
hyU : y ∈ U
hUy : U ∈ 𝓝 y
V : Set (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)
hV :
  Membership.mem (γ :=
    Set (Set (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
    {s : Set (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u) |
      IsClopen s}
    V
hyV : y ∈ V
hVU : V ⊆ U
mp mp₁ mp₂ mp₃ mp₄ : False
mp₅ :
  let Z := of (ULift.{u, 0} (Fin 2));
  let g : Y ⟶ Z :=
    ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
      {
        toFun :=
          (⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen (U := V) hV)) :
            (a : (↑Y.toTop : Type u)) → ULift.{u, 0} (Fin 2)),
        continuous_toFun := ⋯ };
  let h : Y ⟶ Z :=
    ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
      { toFun := fun (x : (↑Y.1 : Type u)) ↦ { down := 1 }, continuous_toFun := ⋯ };
  ∀ (H : h = g), False
mp₆ :
  let Z := of (ULift.{u, 0} (Fin 2));
  let g : Y ⟶ Z :=
    ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
      {
        toFun :=
          (⇑(LocallyConstant.map ULift.up (LocallyConstant.ofIsClopen (U := V) hV)) :
            (a : (↑Y.toTop : Type u)) → ULift.{u, 0} (Fin 2)),
        continuous_toFun := ⋯ };
  let h : Y ⟶ Z :=
    ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
      { toFun := fun (x : (↑Y.1 : Type u)) ↦ { down := 1 }, continuous_toFun := ⋯ };
  ∀
    (H :
      Eq (α := Fin 2)
        (ULift.down.{u, 0}
          ((ConcreteCategory.hom (FC :=
                fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
                (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) h :
              (a :
                  (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦
                      (↑X.toTop : Type u))
                    Y) →
                (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                  Z)
            y))
        (ULift.down.{u, 0}
          ((ConcreteCategory.hom (FC :=
                fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
                (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) g :
              (a :
                  (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦
                      (↑X.toTop : Type u))
                    Y) →
                (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                  Z)
            y))),
    False
mp₇ :
  let Z := of (ULift.{u, 0} (Fin 2));
  let h : Y ⟶ Z :=
    ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
      { toFun := fun (x : (↑Y.1 : Type u)) ↦ { down := 1 }, continuous_toFun := ⋯ };
  ∀
    (H :
      Eq (α := Fin 2)
        (ULift.down.{u, 0}
          ((ConcreteCategory.hom (FC :=
                fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
                (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) h :
              (a : (↑Y.toTop : Type u)) → (↑Z.toTop : Type u))
            y))
        (ULift.down.{u, 0}
          (({
                toFun :=
                  ULift.up ∘
                    fun
                      (x :
                        (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                          Type u)) ↦
                    if x ∈ V then 0 else 1,
                continuous_toFun := ⋯ } :
              (a : (↑Y.toTop : Type u)) → (↑Z.toTop : Type u))
            y))),
    False
mp₈ :
  let Z := of (ULift.{u, 0} (Fin 2));
  let h : Y ⟶ Z :=
    ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))
      { toFun := fun (x : (↑Y.1 : Type u)) ↦ { down := 1 }, continuous_toFun := ⋯ };
  ∀
    (H :
      ULift.down.{u, 0}
          ((ConcreteCategory.hom (FC :=
                fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
                (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)) h).toFun
            y) =
        ULift.down.{u, 0} { down := 0 }),
    False
⊢ False

Tactic:
classical
      let Z := of (ULift.{u} <| Fin 2)
      let g : Y ⟶ Z := ofHom _
        ⟨(LocallyConstant.ofIsClopen hV).map ULift.up, LocallyConstant.continuous _⟩
      let h : Y ⟶ Z := ofHom _ ⟨fun _ => ⟨1⟩, continuous_const⟩
      have H : h = g := by
        rw [← cancel_epi f]
        ext x
        dsimp [g, LocallyConstant.ofIsClopen]
        rw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, ConcreteCategory.hom_ofHom,
          ContinuousMap.coe_mk, Function.comp_apply, if_neg]
        refine mt (fun α => hVU α) ?_
        simp [U, C]
      apply_fun fun e => (e y).down at H
      dsimp [g, LocallyConstant.ofIsClopen] at H
      rw [ContinuousMap.coe_mk, ContinuousMap.coe_mk, Function.comp_apply, if_pos hyV] at H
      exact top_ne_bot H

Premises used: if_pos, ContinuousMap.coe_mk, Function.comp_apply, LocallyConstant.ofIsClopen