-- Module: Mathlib.Geometry.Manifold.VectorField.Pullback
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
H' : Type u_5
inst_6 : TopologicalSpace H'
E' : Type u_6
inst_7 : NormedAddCommGroup E'
inst_8 : NormedSpace 𝕜 E'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst_9 : TopologicalSpace M'
inst_10 : ChartedSpace H' M'
f : M → M'
s : Set M
x₀ : M
V : (x : M') → TangentSpace I' x
m n : ℕ∞ω
t : Set M'
inst_11 : CompleteSpace E
inst_12 : IsManifold I 1 M
inst_13 : IsManifold I' 1 M'
hV :
  ContMDiffWithinAt I' (I'.prod 𝓘(𝕜, E')) (M' := Bundle.TotalSpace.{u_7, u_6, u_6} (B := M') E' (TangentSpace I')) m
    (fun (x : M') ↦ ⟨x, V x⟩) t (f x₀)
hf : ContMDiffWithinAt I I' n f s x₀
hf' : IsInvertible (M := TangentSpace I x₀) (M₂ := TangentSpace I' (f x₀)) (mfderivWithin I I' f s x₀)
hx₀ : x₀ ∈ s
hs : UniqueMDiffOn I s
hmn : m + 1 ≤ n
b₁ : M → M' := f
b₂ : M → M := id
v : (x : M) → TangentSpace I' (f x) := V ∘ f
ϕ : (x : M) → ContinuousLinearMap.{u_1, u_1, u_6, u_3} (RingHom.id 𝕜) (TangentSpace I' (f x)) (TangentSpace I x) :=
  fun (x : M) ↦ inverse (mfderivWithin I I' f s x)
hv :
  ContMDiffWithinAt I I'.tangent (M' := Bundle.TotalSpace.{u_7, u_6, u_6} (B := M') E' (TangentSpace I')) m
    (fun (x : M) ↦ ⟨f x, v x⟩) (s ∩ f ⁻¹' t) x₀
this_1 :
  ContMDiffWithinAt I (E' := E →L[𝕜] E') 𝓘(𝕜, E →L[𝕜] E') (M' := E →L[𝕜] E') m
    (fun (x : M) ↦ inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderivWithin I I' f s x)) s
    x₀
this :
  ContMDiffWithinAt I (E' := E' →L[𝕜] E) 𝓘(𝕜, E' →L[𝕜] E) (M' := E' →L[𝕜] E) m
    (inverse ∘ fun (x : M) ↦
      inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderivWithin I I' f s x))
    s x₀
A :
  Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
      (trivializationAt E (TangentSpace I) x₀) ∈
    𝓝[s] x₀
B :
  f ⁻¹'
      Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_7, u_6, u_6} (B := M') E' (TangentSpace I'))
        (trivializationAt E' (TangentSpace I') (f x₀)) ∈
    𝓝[s] x₀
h :
  ∀ (x : M)
    (hx :
      x ∈
        Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
          (trivializationAt E (TangentSpace I) x₀))
    (h'x :
      x ∈
        f ⁻¹'
          Bundle.Trivialization.baseSet (Z := Bundle.TotalSpace.{u_7, u_6, u_6} (B := M') E' (TangentSpace I'))
            (trivializationAt E' (TangentSpace I') (f x₀))),
    Eq (α := E' →L[𝕜] E) (inCoordinates E' (TangentSpace I') E (TangentSpace I) (b₁ x₀) (b₁ x) (b₂ x₀) (b₂ x) (ϕ x))
      ((inverse ∘ fun (x : M) ↦
          inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderivWithin I I' f s x))
        x)
⊢ (𝓝[s] x₀).EventuallyEq (β := E' →L[𝕜] E)
    (fun (x : M) ↦ inCoordinates E' (TangentSpace I') E (TangentSpace I) (b₁ x₀) (b₁ x) (b₂ x₀) (b₂ x) (ϕ x))
    (inverse ∘ fun (x : M) ↦
      inCoordinates E (TangentSpace I) E' (TangentSpace I') x₀ x (f x₀) (f x) (mfderivWithin I I' f s x))

Tactic:
filter_upwards [A, B] with x hx h'x