specialize h <| by simpa [Set.preimage_val_image_val_eq_self]
  using (show N.E ↓∩ J ⊆ _ from preimage_mono (f := Subtype.val) hJX)
case refine_1
α : Type u_1
β : Type u_2
M : Matroid α
N : Matroid β
e : M ≂ N
I X : Set ↑M.E
hI : M.Indep ↑I
hIX : I ⊆ X
J : Set β
hJ : N.Indep J
hJX : J ⊆ ↑(⇑e '' X)
hIJ : I ⊆ ⇑e ⁻¹' (Subtype.val ⁻¹' J)
h : N.E ↓∩ J ⊆ ⇑e '' X → I ⊆ ⇑e ⁻¹' N.E ↓∩ J → I = ⇑e ⁻¹' N.E ↓∩ J
⊢ ↑(⇑e '' I) = J
case refine_1
α : Type u_1
β : Type u_2
M : Matroid α
N : Matroid β
e : M ≂ N
I X : Set ↑M.E
hI : M.Indep ↑I
hIX : I ⊆ X
J : Set β
hJ : N.Indep J
hJX : J ⊆ ↑(⇑e '' X)
hIJ : I ⊆ ⇑e ⁻¹' (Subtype.val ⁻¹' J)
h : I ⊆ ⇑e ⁻¹' N.E ↓∩ J → I = ⇑e ⁻¹' N.E ↓∩ J
⊢ ↑(⇑e '' I) = J