-- Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
EM : Type u_1
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_2
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace HM M
EN : Type u_3
inst_6 : NormedAddCommGroup EN
inst_7 : NormedSpace 𝕜 EN
HN : Type u_4
inst_8 : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst_9 : TopologicalSpace N
inst_10 : ChartedSpace HN N
EP : Type u_5
inst_11 : NormedAddCommGroup EP
inst_12 : NormedSpace 𝕜 EP
HP : Type u_6
inst_13 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_14 : TopologicalSpace P
inst_15 : ChartedSpace HP P
U : Opens M
V : Opens (↑(↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat) : Type u)
rw :
  Function.Bijective (α :=
    (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj
          (Opposite.op
            (Functor.obj
              (⋯.functor (f :=
                PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
              V))) :
      Type u))
    (β :=
    (↑((Functor.obj
              (TopCat.Presheaf.pushforward CommRingCat
                (PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
              (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
          (Opposite.op
            (Functor.obj
              (⋯.functor (f :=
                PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
              V))) :
      Type u))
    (⇑(ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) (X :=
          (ChartedSpace.locallyRingedSpace IM M).presheaf.obj
            (Opposite.op
              (Functor.obj
                (⋯.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                V)))
          (Y :=
          (Functor.obj
                (TopCat.Presheaf.pushforward CommRingCat
                  (PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
            (Opposite.op
              (Functor.obj
                (⋯.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                V)))
          (NatTrans.app
            (PresheafedSpace.Hom.c
              (InducedCategory.Hom.hom
                (LocallyRingedSpace.Hom.toShHom
                  (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯))))
            (Opposite.op
              (Functor.obj
                (⋯.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                V)))) :
      (a : (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj (Opposite.op (⋯.functor.obj V))) : Type u)) →
        (↑(((TopCat.Presheaf.pushforward CommRingCat
                      (LocallyRingedSpace.Hom.toShHom (ChartedSpace.locallyRingedSpaceMap Subtype.val ⋯)).hom.base).obj
                  (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
              (Opposite.op (⋯.functor.obj V))) :
          Type u))
⊢ IsIso (X :=
    (ChartedSpace.locallyRingedSpace IM M).presheaf.obj
      (Opposite.op
        (Functor.obj
          (⋯.functor (f :=
            PresheafedSpace.Hom.base
              (InducedCategory.Hom.hom
                (LocallyRingedSpace.Hom.toShHom
                  (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
          V)))
    (Y :=
    (Functor.obj
          (TopCat.Presheaf.pushforward CommRingCat
            (PresheafedSpace.Hom.base
              (InducedCategory.Hom.hom
                (LocallyRingedSpace.Hom.toShHom
                  (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
          (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
      (Opposite.op
        (Functor.obj
          (⋯.functor (f :=
            PresheafedSpace.Hom.base
              (InducedCategory.Hom.hom
                (LocallyRingedSpace.Hom.toShHom
                  (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
          V)))
    (NatTrans.app
      (PresheafedSpace.Hom.c
        (InducedCategory.Hom.hom
          (LocallyRingedSpace.Hom.toShHom
            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯))))
      (Opposite.op
        (Functor.obj
          (⋯.functor (f :=
            PresheafedSpace.Hom.base
              (InducedCategory.Hom.hom
                (LocallyRingedSpace.Hom.toShHom
                  (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
          V)))

Tactic:
rw [ConcreteCategory.isIso_iff_bijective]

Premises used: CategoryTheory.ConcreteCategory.isIso_iff_bijective