Theorem: TopCat.Presheaf.stalkToFiber_injective
File: Mathlib/Topology/Sheaves/Sheafify.lean

tactic: apply TopCat.stalkToFiber_injective
state_before: X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ Function.Injective (F.stalkToFiber x)
state_after: case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → F.stalk ↑y),
    (Sheafify.isLocallyGerm F).pred fU →
      ∀ (fV : (y : ↥V.obj) → F.stalk ↑y),
        (Sheafify.isLocallyGerm F).pred fV →
          fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ →
            ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)

tactic: intro U V fU hU fV hV e
state_before: case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
⊢ ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → F.stalk ↑y),
    (Sheafify.isLocallyGerm F).pred fU →
      ∀ (fV : (y : ↥V.obj) → F.stalk ↑y),
        (Sheafify.isLocallyGerm F).pred fV →
          fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ →
            ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
state_after: case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
state_before: case w
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
state_before: case w.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: have wUx := wU ⟨x, mU⟩
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wUx : (fun x_1 => fU ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨x, mU⟩ = F.germ ⟨x, mU⟩ gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: dsimp at wUx
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wUx : (fun x_1 => fU ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨x, mU⟩ = F.germ ⟨x, mU⟩ gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wUx : fU ⟨x, ⋯⟩ = F.germ ⟨x, mU⟩ gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: erw [wUx] at e
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wUx : fU ⟨x, ⋯⟩ = F.germ ⟨x, mU⟩ gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wUx : fU ⟨x, ⋯⟩ = F.germ ⟨x, mU⟩ gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: clear wUx
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wUx : fU ⟨x, ⋯⟩ = F.germ ⟨x, mU⟩ gU
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: have wVx := wV ⟨x, mV⟩
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wVx : (fun x_1 => fV ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨x, mV⟩ = F.germ ⟨x, mV⟩ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: dsimp at wVx
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wVx : (fun x_1 => fV ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨x, mV⟩ = F.germ ⟨x, mV⟩ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wVx : fV ⟨x, ⋯⟩ = F.germ ⟨x, mV⟩ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: erw [wVx] at e
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
e : F.germ ⟨x, mU⟩ gU = fV ⟨x, ⋯⟩
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wVx : fV ⟨x, ⋯⟩ = F.germ ⟨x, mV⟩ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wVx : fV ⟨x, ⋯⟩ = F.germ ⟨x, mV⟩ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: clear wVx
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
wVx : fV ⟨x, ⋯⟩ = F.germ ⟨x, mV⟩ gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: rcases F.germ_eq x mU mV gU gV e with ⟨W, mW, iU', iV', (e' : F.map iU'.op gU = F.map iV'.op gV)⟩
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case w.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: use ⟨W ⊓ (U' ⊓ V'), ⟨mW, mU, mV⟩⟩
state_before: case w.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∃ iU iV,
    ∀ (w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: refine ⟨?_, ?_, ?_⟩
state_before: case h
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∃ iU iV,
    ∀ (w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case h.refine_1
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ { obj := W ⊓ (U' ⊓ V'), property := ⋯ } ⟶ U

case h.refine_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ { obj := W ⊓ (U' ⊓ V'), property := ⋯ } ⟶ V

case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∀ (w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: change W ⊓ (U' ⊓ V') ⟶ U.obj
state_before: case h.refine_1
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ { obj := W ⊓ (U' ⊓ V'), property := ⋯ } ⟶ U
state_after: case h.refine_1
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ W ⊓ (U' ⊓ V') ⟶ U.obj

tactic: exact Opens.infLERight _ _ ≫ Opens.infLELeft _ _ ≫ iU
state_before: case h.refine_1
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ W ⊓ (U' ⊓ V') ⟶ U.obj
state_after: no goals

tactic: change W ⊓ (U' ⊓ V') ⟶ V.obj
state_before: case h.refine_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ { obj := W ⊓ (U' ⊓ V'), property := ⋯ } ⟶ V
state_after: case h.refine_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ W ⊓ (U' ⊓ V') ⟶ V.obj

tactic: exact Opens.infLERight _ _ ≫ Opens.infLERight _ _ ≫ iV
state_before: case h.refine_2
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ W ⊓ (U' ⊓ V') ⟶ V.obj
state_after: no goals

tactic: intro w
state_before: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
⊢ ∀ (w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
⊢ fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: specialize wU ⟨w.1, w.2.2.1⟩
state_before: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
wU : ∀ (x_1 : ↥U'), (fun x_2 => fU ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gU
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
⊢ fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
wU : (fun x_1 => fU ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gU
⊢ fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: specialize wV ⟨w.1, w.2.2.2⟩
state_before: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
wV : ∀ (x_1 : ↥V'), (fun x_2 => fV ((fun x_3 => ⟨↑x_3, ⋯⟩) x_2)) x_1 = F.germ x_1 gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
wU : (fun x_1 => fU ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gU
⊢ fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
wU : (fun x_1 => fU ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gU
wV : (fun x_1 => fV ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gV
⊢ fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

tactic: dsimp at wU wV ⊢
state_before: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
wU : (fun x_1 => fU ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gU
wV : (fun x_1 => fV ((fun x_2 => ⟨↑x_2, ⋯⟩) x_1)) ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gV
⊢ fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)
state_after: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
wU : fU ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gU
wV : fV ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gV
⊢ fU ⟨↑w, ⋯⟩ = fV ⟨↑w, ⋯⟩

tactic: erw [wU, ← F.germ_res iU' ⟨w, w.2.1⟩, wV, ← F.germ_res iV' ⟨w, w.2.1⟩,
  CategoryTheory.types_comp_apply, CategoryTheory.types_comp_apply, e']
state_before: case h.refine_3
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U' ⟶ U.obj
gU : F.obj { unop := U' }
V' : Opens ↑X
mV : ↑⟨x, ⋯⟩ ∈ V'
iV : V' ⟶ V.obj
gV : F.obj { unop := V' }
e : F.germ ⟨x, mU⟩ gU = F.germ ⟨x, mV⟩ gV
W : Opens ↑X
mW : x ∈ W
iU' : W ⟶ U'
iV' : W ⟶ V'
e' : F.map iU'.op gU = F.map iV'.op gV
w : ↥{ obj := W ⊓ (U' ⊓ V'), property := ⋯ }.obj
wU : fU ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gU
wV : fV ⟨↑w, ⋯⟩ = F.germ ⟨↑w, ⋯⟩ gV
⊢ fU ⟨↑w, ⋯⟩ = fV ⟨↑w, ⋯⟩
state_after: no goals