rw [← (he i).basis_iff, ← (he i).inter_eq_of_indep hB.indep]
case 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
he : ∀ (i : ι), M.LoopEquiv (Xs i) (Ys i)
B : Set α
hB : M.ModularBase B Xs
i : ι
⊢ M.Basis (Ys i ∩ B) (Ys i)
case 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
he : ∀ (i : ι), M.LoopEquiv (Xs i) (Ys i)
B : Set α
hB : M.ModularBase B Xs
i : ι
⊢ M.Basis (Xs i ∩ B) (Xs i)