rw [inter_eq_self_of_subset_left <| iUnion_subset_iff.1 hIB i]
case 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
h : M.Indep (⋃ i, Xs i)
B : Set α
hB : M.Base B
hIB : ⋃ i, Xs i ⊆ B
i : ι
⊢ M.Basis (Xs i ∩ B) (Xs i)
case 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
h : M.Indep (⋃ i, Xs i)
B : Set α
hB : M.Base B
hIB : ⋃ i, Xs i ⊆ B
i : ι
⊢ M.Basis (Xs i) (Xs i)