rw [map_base_iff] at hB
case refine_1.intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs✝ Ys Js Is : ι → Set α
i j : ι
e f✝ : α
β : Type u_3
f : α → β
hf : InjOn f M.E
Xs : ι → Set β
B : Set β
hB : (M.map f hf).Base B
h : ∀ (i : ι), ∃ I₀ X₀, M.Basis I₀ X₀ ∧ Xs i ∩ B = f '' I₀ ∧ Xs i = f '' X₀
⊢ ∃ Ys, M.ModularFamily Ys ∧ ∀ (i : ι), Xs i = f '' Ys i
case refine_1.intro.intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs✝ Ys Js Is : ι → Set α
i j : ι
e f✝ : α
β : Type u_3
f : α → β
hf : InjOn f M.E
Xs : ι → Set β
B : Set β
hB : ∃ B₀, M.Base B₀ ∧ B = f '' B₀
h : ∀ (i : ι), ∃ I₀ X₀, M.Basis I₀ X₀ ∧ Xs i ∩ B = f '' I₀ ∧ Xs i = f '' X₀
⊢ ∃ Ys, M.ModularFamily Ys ∧ ∀ (i : ι), Xs i = f '' Ys i