simp only [image_symm_eq_preimage, e.indep_image_iff, image_preimage,
  Subtype.image_preimage_coe, inter_eq_self_of_subset_right hJ.subset_ground, hJ,
  image_subset_iff, preimage_val_image_val_eq_self, preimage_subset_iff, image_val_inj,
  true_implies] at h
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 :
  M.Indep ↑(⇑e.symm '' N.E ↓∩ J) →
    ↑(⇑e.symm '' N.E ↓∩ J) ⊆ ↑X → I ⊆ Subtype.val ⁻¹' ↑(⇑e.symm '' N.E ↓∩ J) → ↑I = ↑(⇑e.symm '' 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 : N.E ↓∩ J ⊆ ⇑e '' X → I ⊆ ⇑e ⁻¹' N.E ↓∩ J → I = ⇑e ⁻¹' N.E ↓∩ J
⊢ ↑(⇑e '' I) = J