-- Module: Mathlib.Condensed.Discrete.LocallyConstant
Goal:
P : TopCat.{u} → Prop
inst : ∀ (S : CompHausLike P) (p : (↑(S.toTop (P := P)) : Type u) → Prop), HasProp P (Subtype p)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst_1 : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑(S.toTop (P := P)) : Type u) (Y.obj (op (of P PUnit.{u + 1})))
T : CompHausLike P
g : T ⟶ S
X_1 : TopCat
inst_2 : HasExplicitFiniteCoproducts P
inst_3 : HasExplicitPullbacks P
hs :
  ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y),
    EffectiveEpi f →
      Surjective (α := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X) (β :=
        (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) Y)
        (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
              C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC := fun (X : CompHausLike P) ↦
              (↑(X.toTop (P := P)) : Type u)) f) :
          (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) Y)
inst_4 : HasExplicitFiniteCoproducts P
X : Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))
x : X.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P))).obj (op (of P PUnit.{u + 1}))
this_1 : Preregular.{u, u + 1} (CompHausLike P)
this : PreservesFiniteProducts (Functor.obj (sheafToPresheaf (coherentTopology (CompHausLike P)) (Type (max u w))) X) :=
  ⋯
a :
  Fiber (Y := (↑((of P PUnit.{u + 1}).toTop (P := P)) : Type u)) (Z :=
    X.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P))).obj (op (of P PUnit.{u + 1})))
    (⇑((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app (unit P hs)
                  (X.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P))).obj (op (of P PUnit.{u + 1})))) :
              (a : (fun (X : Type (max u w)) ↦ X) ((𝟭 (Type (max u w))).obj (X.obj.obj (op (of P PUnit.{u + 1}))))) →
                (fun (X : Type (max u w)) ↦ X)
                  ((functor P hs ⋙
                        (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w))).obj
                          (op (of P PUnit.{u + 1}))).obj
                    (X.obj.obj (op (of P PUnit.{u + 1})))))
            x :
          (fun (X : Type (max u w)) ↦ X)
            ((Functor.comp (D := Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                  (functor.{u, w} P hs :
                    Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                  (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                      (op (of P PUnit.{u + 1})) :
                    Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)) ⥤
                      Type (max u w))).obj
              (X.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P))).obj (op (of P PUnit.{u + 1}))))) :
      (a : (↑(of P PUnit.{u + 1}).toTop : Type u)) → X.obj.obj (op (of P PUnit.{u + 1})))
⊢ PUnit.unit ∈ (↑a : Set (↑(of P PUnit.{u + 1}).toTop : Type u))

Tactic:
simp [mem_iff_eq_image, ← map_preimage_eq_image]

Premises used: Function.Fiber.mem_iff_eq_image, Function.Fiber.map_preimage_eq_image