/-- The `stalkToFiber` map is injective at `x` if any two allowed sections which agree at `x`
agree on some neighborhood of `x`.
-/
theorem stalkToFiber_injective (P : LocalPredicate T) (x : X)
    (w :
      ∀ (U V : OpenNhds x) (fU : ∀ y : U.1, T y) (_ : P.pred fU) (fV : ∀ y : V.1, T y)
        (_ : P.pred fV) (_ : fU ⟨x, U.2⟩ = fV ⟨x, V.2⟩),
        ∃ (W : OpenNhds x) (iU : W ⟶ U) (iV : W ⟶ V), ∀ w : W.1,
          fU (iU w : U.1) = fV (iV w : V.1)) :
    Function.Injective (stalkToFiber P x) := fun tU tV h => by
  -- We promise to provide all the ingredients of the proof later:
  let Q :
    ∃ (W : (OpenNhds x)ᵒᵖ) (s : ∀ w : (unop W).1, T w) (hW : P.pred s),
      tU = (subsheafToTypes P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ ∧
        tV = (subsheafToTypes P).presheaf.germ ⟨x, (unop W).2⟩ ⟨s, hW⟩ :=
    ?_
  · choose W s hW e using Q
    exact e.1.trans e.2.symm
  -- Then use induction to pick particular representatives of `tU tV : stalk x`
  obtain ⟨U, ⟨fU, hU⟩, rfl⟩ := jointly_surjective'.{v, v} tU
  obtain ⟨V, ⟨fV, hV⟩, rfl⟩ := jointly_surjective'.{v, v} tV
  · -- Decompose everything into its constituent parts:
    dsimp
    simp only [stalkToFiber, Types.Colimit.ι_desc_apply'] at h
    specialize w (unop U) (unop V) fU hU fV hV h
    rcases w with ⟨W, iU, iV, w⟩
    -- and put it back together again in the correct order.
    refine' ⟨op W, fun w => fU (iU w : (unop U).1), P.res _ _ hU, _⟩
    

Proof state:
case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_1
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥((OpenNhds.inclusion x).op.obj U).unop) → T ↑x_1
hU : PrelocalPredicate.pred P.toPrelocalPredicate fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥((OpenNhds.inclusion x).op.obj V).unop) → T ↑x_1
hV : PrelocalPredicate.pred P.toPrelocalPredicate fV
h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) }
W : OpenNhds x
iU : W ⟶ U.unop
iV : W ⟶ V.unop
w :
  ∀ (w : ↥W.obj),
    fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) =
      fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w)
⊢ (op W).unop.obj ⟶ U.unop.obj
case refine_1.intro.intro.mk.intro.intro.mk.intro.intro.intro.refine'_2
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥((OpenNhds.inclusion x).op.obj U).unop) → T ↑x_1
hU : PrelocalPredicate.pred P.toPrelocalPredicate fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥((OpenNhds.inclusion x).op.obj V).unop) → T ↑x_1
hV : PrelocalPredicate.pred P.toPrelocalPredicate fV
h : fU { val := x, property := (_ : x ∈ U.unop.obj) } = fV { val := x, property := (_ : x ∈ V.unop.obj) }
W : OpenNhds x
iU : W ⟶ U.unop
iV : W ⟶ V.unop
w :
  ∀ (w : ↥W.obj),
    fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w) =
      fV ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑V.unop.obj) }) w)
⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =
      Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) }
        { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w),
          property :=
            (_ :
              PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 =>
                fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) } ∧
    colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V { val := fV, property := hV } =
      Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ (op W).unop.obj) }
        { val := fun w => fU ((fun x_1 => { val := ↑x_1, property := (_ : ↑x_1 ∈ ↑U.unop.obj) }) w),
          property :=
            (_ :
              PrelocalPredicate.pred P.toPrelocalPredicate fun x_1 =>
                fU ((fun x_2 => { val := ↑x_2, property := (_ : ↑x_2 ∈ ↑U.unop.obj) }) x_1)) }

Next tactic:
rcases W with ⟨W, m⟩