simp only [image_subset_iff, preimage_val_image_val_eq_self, image_symm_eq_preimage,
  preimage_subset_iff, image_val_inj] at hIC hC
case intro
α : Type u_1
M : Matroid α
C✝ C' I✝ X K C₁ C₂ R : Set α
e✝ f x y : α
β : Type u_2
N : Matroid β
e : M ≂ N
C : Set ↑M.E
I : Set ↑N.E
hI : N.Dep ↑I
hIC : ↑I ⊆ ↑(⇑e '' C)
hC : ↑(⇑e.symm '' I) ⊆ ↑C → ↑(⇑e.symm '' I) = ↑C
⊢ ↑I = ↑(⇑e '' C)
case intro
α : Type u_1
M : Matroid α
C✝ C' I✝ X K C₁ C₂ R : Set α
e✝ f x y : α
β : Type u_2
N : Matroid β
e : M ≂ N
C : Set ↑M.E
I : Set ↑N.E
hI : N.Dep ↑I
hIC : I ⊆ ⇑e '' C
hC : I ⊆ ⇑e '' C → ⇑e ⁻¹' I = C
⊢ ↑I = ↑(⇑e '' C)