-- Mathlib/Topology/Sheaves/Stalks.lean
theorem stalkFunctor_map_injective_of_app_injective {F G : Presheaf C X} (f : F ⟶ G)
    (h : ∀ U : Opens X, Function.Injective (f.app (op U))) (x : X) :
    Function.Injective ((stalkFunctor C x).map f) := fun s t hst => by
  rcases germ_exist F x s with ⟨U₁, hxU₁, s, rfl⟩
  rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩
  rw [stalkFunctor_map_germ_apply, stalkFunctor_map_germ_apply] at hst
  obtain ⟨W, hxW, iWU₁, iWU₂, heq⟩ := G.germ_eq x hxU₁ hxU₂ _ _ hst
  rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply, ← f.naturality, ← f.naturality,
    ConcreteCategory.comp_apply, ConcreteCategory.comp_apply] at heq
  replace heq := h W heq
  convert congr_arg (F.germ _ x hxW) heq using 1
  exacts [(F.germ_res_apply iWU₁ x hxW s).symm, (F.germ_res_apply iWU₂ x hxW t).symm]

Proof state:
case h.e'_2.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(ConcreteCategory.hom (f.app (op U)))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : ToType (F.obj (op U₁))
U₂ : Opens ↑X
hxU₂ : x ∈ U₂
t : ToType (F.obj (op U₂))
hst :
  (ConcreteCategory.hom (G.germ U₁ x hxU₁)) ((ConcreteCategory.hom (f.app (op U₁))) s) =
    (ConcreteCategory.hom (G.germ U₂ x hxU₂)) ((ConcreteCategory.hom (f.app (op U₂))) t)
W : Opens ↑X
hxW : x ∈ W
iWU₁ : W ⟶ U₁
iWU₂ : W ⟶ U₂
heq : (ConcreteCategory.hom (F.map iWU₁.op)) s = (ConcreteCategory.hom (F.map iWU₂.op)) t
e_1✝ : CC ((stalkFunctor C x).obj F) = CC (F.stalk x)
⊢ (ConcreteCategory.hom (F.germ U₁ x hxU₁)) s =
    (ConcreteCategory.hom (F.germ W x hxW)) ((ConcreteCategory.hom (F.map iWU₁.op)) s)

case h.e'_3.h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(ConcreteCategory.hom (f.app (op U)))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : ToType (F.obj (op U₁))
U₂ : Opens ↑X
hxU₂ : x ∈ U₂
t : ToType (F.obj (op U₂))
hst :
  (ConcreteCategory.hom (G.germ U₁ x hxU₁)) ((ConcreteCategory.hom (f.app (op U₁))) s) =
    (ConcreteCategory.hom (G.germ U₂ x hxU₂)) ((ConcreteCategory.hom (f.app (op U₂))) t)
W : Opens ↑X
hxW : x ∈ W
iWU₁ : W ⟶ U₁
iWU₂ : W ⟶ U₂
heq : (ConcreteCategory.hom (F.map iWU₁.op)) s = (ConcreteCategory.hom (F.map iWU₂.op)) t
e_1✝ : CC ((stalkFunctor C x).obj F) = CC (F.stalk x)
⊢ (ConcreteCategory.hom (F.germ U₂ x hxU₂)) t =
    (ConcreteCategory.hom (F.germ W x hxW)) ((ConcreteCategory.hom (F.map iWU₂.op)) t)

Tactic:
exacts [(F.germ_res_apply iWU₁ x hxW s).symm, (F.germ_res_apply iWU₂ x hxW t).symm]

Result:
no goals

Proof state:
case intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(ConcreteCategory.hom (f.app (op U)))
x : ↑X
t : CC ((stalkFunctor C x).obj F)
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : ToType (F.obj (op U₁))
hst :
  (ConcreteCategory.hom ((stalkFunctor C x).map f)) ((ConcreteCategory.hom (F.germ U₁ x hxU₁)) s) =
    (ConcreteCategory.hom ((stalkFunctor C x).map f)) t
⊢ (ConcreteCategory.hom (F.germ U₁ x hxU₁)) s = t

Tactic:
rcases germ_exist F x t with ⟨U₂, hxU₂, t, rfl⟩

Result:
case intro.intro.intro.intro.intro.intro
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝ : PreservesFilteredColimits (forget C)
F G : Presheaf C X
f : F ⟶ G
h : ∀ (U : Opens ↑X), Function.Injective ⇑(ConcreteCategory.hom (f.app (op U)))
x : ↑X
U₁ : Opens ↑X
hxU₁ : x ∈ U₁
s : ToType (F.obj (op U₁))
U₂ : Opens ↑X
hxU₂ : x ∈ U₂
t : ToType (F.obj (op U₂))
hst :
  (ConcreteCategory.hom ((stalkFunctor C x).map f)) ((ConcreteCategory.hom (F.germ U₁ x hxU₁)) s) =
    (ConcreteCategory.hom ((stalkFunctor C x).map f)) ((ConcreteCategory.hom (F.germ U₂ x hxU₂)) t)
⊢ (ConcreteCategory.hom (F.germ U₁ x hxU₁)) s = (ConcreteCategory.hom (F.germ U₂ x hxU₂)) t