-- Lean 4 proof step from Mathlib.Topology.Category.LightProfinite.Basic (goal)
Goal:
X Y : LightProfinite
f : X ⟶ Y
y :
  (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
    Type u)
hy :
  ∀
    (a :
      (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
        Type u)),
    (ConcreteCategory.hom (FC :=
            fun
              (x1 x2 :
                CompHausLike fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
            C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u),
              (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u)))
            (CC :=
            fun
              (X :
                CompHausLike fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
            (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{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) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
      (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
        Type u))
    Y) :=
  Set.range (ι :=
    (fun
          (X :
            CompHausLike fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
        (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
          Type u))
      X)
    (⇑(ConcreteCategory.hom (FC :=
          fun
            (x1 x2 :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u),
            (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u)))
          (CC :=
          fun
            (X :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
            Type u))
          f) :
      (a :
          (fun
                (X :
                  CompHausLike fun (X : TopCat) ↦
                    TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
              (↑X.toTop : Type u))
            X) →
        (fun
              (X :
                CompHausLike fun (X : TopCat) ↦
                  TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑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) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
      (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{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) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)
hV :
  Membership.mem (γ :=
    Set
      (Set
        (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
          Type u)))
    {s :
      Set
        (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{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) ∧ SecondCountableTopology.{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) ∧ SecondCountableTopology.{u} (↑X : Type u))
      { toFun := fun (x : (↑Y.1 : Type u)) ↦ { down := 1 }, continuous_toFun := ⋯ };
  ∀ (H : h = g), False
⊢ False
have mp₆ : let Z := of (ULift.{u, 0} (Fin 2));
let g : Y ⟶ Z :=
  ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{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 := sorry };
let h : Y ⟶ Z :=
  ofHom (fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
    { toFun := fun (x : (↑Y.1 : Type u)) ↦ { down := 1 }, continuous_toFun := sorry };
∀
  (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) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
              C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                  Type u),
                (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                  Type u)))
              (CC :=
              fun
                (X :
                  CompHausLike fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u))
              h :
            (a :
                (fun
                      (X :
                        CompHausLike fun (X : TopCat) ↦
                          TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                    (↑X.toTop : Type u))
                  Y) →
              (fun
                    (X :
                      CompHausLike fun (X : TopCat) ↦
                        TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑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) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
              C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                  Type u),
                (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                  Type u)))
              (CC :=
              fun
                (X :
                  CompHausLike fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u))
              g :
            (a :
                (fun
                      (X :
                        CompHausLike fun (X : TopCat) ↦
                          TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                    (↑X.toTop : Type u))
                  Y) →
              (fun
                    (X :
                      CompHausLike fun (X : TopCat) ↦
                        TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                  (↑X.toTop : Type u))
                Z)
          y))),
  False
Tactic: classical
      let Z := of (ULift.{u} <| Fin 2)
      let g : Y ⟶ Z := CompHausLike.ofHom _
        ⟨(LocallyConstant.ofIsClopen hV).map ULift.up, LocallyConstant.continuous _⟩
      let h : Y ⟶ Z := CompHausLike.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, 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