-- Module: Mathlib.AlgebraicGeometry.Gluing
Goal:
D : GlueData
i j : D.J
x : ↥(D.U i)
y : ↥(D.U j)
rw :
  Eq (α := ↥D.glued) ((D.ι i) x : ↥D.glued) ((D.ι j) y : ↥D.glued) ↔
    Eq (α := ↥D.glued)
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              (D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι i ≫
                D.isoCarrier.inv) :
            (a :
                (↑(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.U i) :
                  Type u)) →
              ↥D.glued)
          x :
        ↥D.glued)
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              D.isoCarrier.inv :
            (a :
                (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued :
                  Type u)) →
              ↥D.glued)
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                  (Y :=
                  D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued)
                  (D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι j) :
                (a :
                    (↑(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.U
                          j) :
                      Type u)) →
                  (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued :
                    Type u))
              y :
            (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued :
              Type u)) :
        ↥D.glued)
rw₁ : Mono D.isoCarrier.inv
⊢ Eq (α := ↥D.glued) ((D.ι i) x : ↥D.glued) ((D.ι j) y : ↥D.glued) ↔
    Eq (α :=
      (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              (Y := D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued)
              (D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι i) :
            (a :
                (↑(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.U i) :
                  Type u)) →
              (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued :
                Type u))
          x :
        (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              (Y := D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued)
              (D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι j) :
            (a :
                (↑(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.U j) :
                  Type u)) →
              (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued :
                Type u))
          y :
        (↑D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.glued : Type u))

Tactic:
rw [← ((TopCat.mono_iff_injective D.isoCarrier.inv).mp _).eq_iff, ← ConcreteCategory.comp_apply]

Premises used: CategoryTheory.ConcreteCategory.comp_apply, AlgebraicGeometry.Scheme.GlueData.isoCarrier, Function.Injective.eq_iff, Iff.mp, TopCat.mono_iff_injective, CategoryTheory.Iso.inv