-- Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
P : Functor.{v_1, u_4, u_1, u_4 + 1} Cᵒᵖ (Type u_4)
hP :
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst_4 : HasPullback π π],
    IsIso (Y :=
      equalizer (Y := P.obj (op (pullback π π))) (P.map (Quiver.Hom.op (pullback.fst π π)))
        (P.map (Quiver.Hom.op (pullback.snd π π))))
      (equalizer.lift (P.map π.op) ⋯)
X B : C
π : X ⟶ B
inst_1 : EffectiveEpi π
inst_2 : HasPullback π π
mpr :
  IsIso (Y :=
    (↑{x : P.obj (op X) |
          Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                  (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                x :
              (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                  (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) → (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                x :
              (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
      Type u_4))
    (equalizer.lift (P.map π.op) ⋯ ≫
      Iso.hom
        (Types.equalizerIso (P.map (Quiver.Hom.op (pullback.fst π π))) (P.map (Quiver.Hom.op (pullback.snd π π)))))
⊢ Function.Bijective (α := (fun (X : Type u_4) ↦ X) (P.obj (op B))) (β :=
    (fun (X : Type u_4) ↦ X)
      (↑{x : P.obj (op X) |
            Eq (α := (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.fst π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) (P.map (Quiver.Hom.op (pullback.snd π π))) :
                    (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                      (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                  x :
                (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))} :
        Type u_4))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) :
      (a : (fun (X : Type u_4) ↦ X) (P.obj (op B))) →
        (fun (X : Type u_4) ↦ X)
          (↑{x : P.obj (op X) |
                (ConcreteCategory.hom (P.map (pullback.fst π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x =
                  (ConcreteCategory.hom (P.map (pullback.snd π π).op) :
                      (a : (fun (X : Type u_4) ↦ X) (P.obj (op X))) →
                        (fun (X : Type u_4) ↦ X) (P.obj (op (pullback π π))))
                    x} :
            Type u_4))

Tactic:
rw [mapToEqualizer_eq_comp, ← isIso_iff_bijective]

Premises used: CategoryTheory.isIso_iff_bijective, CategoryTheory.regularTopology.mapToEqualizer_eq_comp