∀ {α : Type u_1} {β : Type u_2} {f : α → β} {M : Matroid α} {B : Set α}
  (motive : (∃ B₀, M.IsBase B₀ ∧ f '' B = f '' B₀) → Prop) (x : ∃ B₀, M.IsBase B₀ ∧ f '' B = f '' B₀),
  (∀ (J : Set α) (hJ : M.IsBase J) (hIJ : f '' B = f '' J), motive ⋯) → motive x