/-- 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
    

Proof state:
case refine_1.intro.intro.mk.intro.intro.mk
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w :
  ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y),
    PrelocalPredicate.pred P.toPrelocalPredicate fU →
      ∀ (fV : (y : ↥V.obj) → T ↑y),
        PrelocalPredicate.pred P.toPrelocalPredicate fV →
          fU { val := x, property := (_ : x ∈ U.obj) } = fV { val := x, property := (_ : x ∈ V.obj) } →
            ∃ W iU iV,
              ∀ (w : ↥W.obj),
                fU ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑U.obj) }) w) =
                  fV ((fun x_4 => { val := ↑x_4, property := (_ : ↑x_4 ∈ ↑V.obj) }) w)
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 :
  stalkToFiber P x
      (colimit.ι
        (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj
          (Sheaf.presheaf (subsheafToTypes P)))
        U { val := fU, property := hU }) =
    stalkToFiber P x
      (colimit.ι
        (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj
          (Sheaf.presheaf (subsheafToTypes P)))
        V { val := fV, property := hV })
⊢ ∃ W s,
    ∃ (hW : PrelocalPredicate.pred P.toPrelocalPredicate s),
      colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U { val := fU, property := hU } =
          Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) }
            { val := s, property := hW } ∧
        colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V
            { val := fV, property := hV } =
          Presheaf.germ (subpresheafToTypes P.toPrelocalPredicate) { val := x, property := (_ : x ∈ W.unop.obj) }
            { val := s, property := hW }

Next tactic:
simp only [stalkToFiber, Types.Colimit.ι_desc_apply'] at h