simp only [← e.indep_image_iff, Subtype.image_preimage_coe,
  inter_eq_self_of_subset_right hJ.subset_ground, hJ, image_subset_iff,
  preimage_val_image_val_eq_self, preimage_image, image_val_inj, true_implies] at h
case refine_2
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 α
hJ : M.Indep J
hJX : J ⊆ ↑X
hIJ : I ⊆ Subtype.val ⁻¹' J
h :
  N.Indep ↑(⇑e '' M.E ↓∩ J) →
    ↑(⇑e '' M.E ↓∩ J) ⊆ ↑(⇑e '' X) → I ⊆ ⇑e ⁻¹' (Subtype.val ⁻¹' ↑(⇑e '' M.E ↓∩ J)) → ↑(⇑e '' I) = ↑(⇑e '' M.E ↓∩ J)
⊢ ↑I = J
case refine_2
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 α
hJ : M.Indep J
hJX : J ⊆ ↑X
hIJ : I ⊆ Subtype.val ⁻¹' J
h : M.E ↓∩ J ⊆ X → I ⊆ M.E ↓∩ J → ⇑e '' I = ⇑e '' M.E ↓∩ J
⊢ ↑I = J