awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
left :
  (Opens.map (D.f j k).base).obj
      (unop
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι j)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v))))) =
    Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v))
⊢ op
      ((Opens.map (D.f j k).base).obj
        (unop
          (op
            (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
              (Opens.map (Hom.base (D.ι j)) :
                Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
              (Functor.obj
                  (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                  U :
                Opens (↑(↑D.glued : TopCat) : Type v)))))) =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))

Tactic:
congr 1
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
left :
  (↑((Opens.map (D.f j k).base).obj
          (unop
            (op
              (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                (Opens.map (Hom.base (D.ι j)) :
                  Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
                (Functor.obj
                    (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                      Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                    U :
                  Opens (↑(↑D.glued : TopCat) : Type v)))))) :
      Set (↑(↑(D.V (j, k)) : TopCat) : Type v)) =
    (↑(Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
              Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
                Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)) :
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v)) :
      Set (↑(↑(D.V (j, k)) : TopCat) : Type v))
⊢ (Opens.map (D.f j k).base).obj
      (unop
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι j)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v))))) =
    Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v))

Tactic:
ext1
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
left :
  Eq (α := (D.V (j, k)).Hom D.glued) (D.f j k ≫ D.ι j) (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k)))
⊢ Eq (α := Set (↑(↑(D.V (j, k)) : TopCat) : Type v))
    (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (↑D.glued : TopCat)) (Hom.base (D.f j k ≫ D.ι j))) :
        (a : (↑(↑(D.V (j, k)) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.preimage (β := (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(D.diagram.multispan.obj (WalkingMultispan.left (j, k))) : TopCat)) (Y :=
            (↑(Limits.colimit D.diagram.multispan) : TopCat))
            (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k))))) :
        (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.left (j, k))) : TopCat) : Type v)) →
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))

Tactic:
congr 4
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
⊢ Eq (α := (D.V (j, k)).Hom D.glued) (D.f j k ≫ D.ι j) (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k)))

Tactic:
colimit.w 𝖣.diagram.multispan (WalkingMultispan.Hom.fst (j, k))

Premises used: CategoryTheory.GlueData.diagram, CategoryTheory.Limits.colimit, CategoryTheory.GlueData.ι, CategoryTheory.Limits.WalkingMultispan.Hom.fst, CategoryTheory.Limits.MultispanIndex.multispan, CategoryTheory.Limits.colimit.w, CategoryTheory.GlueData.glued, AlgebraicGeometry.PresheafedSpace.GlueData.toGlueData, CategoryTheory.Limits.colimit.ι
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
⊢ Eq (α := (D.V (j, k)).Hom D.glued) (D.f j k ≫ D.ι j) (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k)))

Tactic:
exact colimit.w 𝖣.diagram.multispan (WalkingMultispan.Hom.fst (j, k))
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j : (MultispanShape.prod D.J).R
⊢ (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.right j))

Tactic:
exact D.opensImagePreimageMap i j U

Premises used: AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
w :
  op U =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))
h :
  Eq (α := (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.right i)))
    (D.ιInvApp U ≫ D.diagramOverOpenπ U i) ((D.U i).presheaf.map (eqToHom ⋯))
⊢ ∃ (eq :
    op U =
      op
        (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
          (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))),
    D.ιInvApp U ≫ D.diagramOverOpenπ U i = (D.U i).presheaf.map (eqToHom eq)

Tactic:
fconstructor
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
w :
  U =
    Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v))
⊢ op U =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))

Tactic:
congr
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
w :
  (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) =
    (↑(Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
              Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
                Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)) :
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) :
      Set (↑(↑(D.U i) : TopCat) : Type v))
⊢ U =
    Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v))

Tactic:
ext1
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
w :
  ∀ (x : (↑(↑(D.U i) : TopCat) : Type v)),
    x ∈ (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) ↔
      Membership.mem (γ := Set (↑(↑(D.U i) : TopCat) : Type v))
        (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC :=
                TopCat.carrier) (X := (↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat))
                (Y := (↑(Limits.colimit D.diagram.multispan) : TopCat))
                (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i)))))) :
            (a : (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) →
              (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
          ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC :=
                  TopCat.carrier) (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
              (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
            (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
        x
⊢ (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) =
    Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat)) (Y :=
            (↑(Limits.colimit D.diagram.multispan) : TopCat))
            (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i)))))) :
        (a : (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) →
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))

Tactic:
ext1 x
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
x : (↑(↑(D.U i) : TopCat) : Type v)
w :
  x ∈ U ↔
    ∃ x_1 ∈ U,
      Eq (α := (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i)) :
              (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
            x_1 :
          (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (X := (↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat)) (Y :=
                (↑(Limits.colimit D.diagram.multispan) : TopCat))
                (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.right i))) :
              (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat) : Type v)) →
                (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            x :
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
⊢ x ∈ (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) ↔
    Membership.mem (γ := Set (↑(↑(D.U i) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (X := (↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat)) (Y :=
              (↑(Limits.colimit D.diagram.multispan) : TopCat))
              (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i)))))) :
          (a : (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) →
            (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
            (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
      x

Tactic:
simp only [SetLike.mem_coe, unop_op, Set.mem_preimage, Set.mem_image]

Premises used: Set.mem_image, SetLike.mem_coe, Opposite.unop_op, Set.mem_preimage
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
x : (↑(↑(D.U i) : TopCat) : Type v)
⊢ x ∈ U ↔
    ∃ x_1 ∈ U,
      Eq (α := (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i)) :
              (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
            x_1 :
          (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (X := (↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat)) (Y :=
                (↑(Limits.colimit D.diagram.multispan) : TopCat))
                (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.right i))) :
              (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat) : Type v)) →
                (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            x :
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))

Tactic:
⟨fun h => ⟨_, h, rfl⟩, ?_⟩

Premises used: And.intro, Iff.intro, Exists.intro, rfl
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
x : (↑(↑(D.U i) : TopCat) : Type v)
w :
  (∃ x_1 ∈ U,
      Eq (α := (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i)) :
              (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
            x_1 :
          (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (X := (↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat)) (Y :=
                (↑(Limits.colimit D.diagram.multispan) : TopCat))
                (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.right i))) :
              (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat) : Type v)) →
                (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            x :
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))) →
    x ∈ U
⊢ x ∈ U ↔
    ∃ x_1 ∈ U,
      Eq (α := (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i)) :
              (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
            x_1 :
          (↑(↑D.glued : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (X := (↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat)) (Y :=
                (↑(Limits.colimit D.diagram.multispan) : TopCat))
                (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.right i))) :
              (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat) : Type v)) →
                (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
            x :
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))

Tactic:
refine ⟨fun h => ⟨_, h, rfl⟩, ?_⟩
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
x y : (↑(↑(D.U i) : TopCat) : Type v)
h1 : y ∈ U
h2 :
  Eq (α := (↑(↑D.glued : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (Y :=
            (↑D.glued : TopCat)) (Hom.base (D.ι i)) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
        y :
      (↑(↑D.glued : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat)) (Y :=
            (↑(Limits.colimit D.diagram.multispan) : TopCat))
            (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.right i))) :
          (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.right i)) : TopCat) : Type v)) →
            (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        x :
      (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
⊢ y ∈ U

Tactic:
h1