-- Module: Mathlib.Topology.Category.LightProfinite.Basic
Goal:
X Y : LightProfinite
f : X ⟶ Y
mp :
  (∃ (b :
      (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
        Type u)),
      ∀
        (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 ≠
          b) →
    ¬Epi f
⊢ Epi f →
    ∀
      (b :
        (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
              TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
          Type u)),
      ∃ (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 =
          b

Tactic:
contrapose!