intros.
unfold indexed_intersection in H.
simpl in H.
apply H.
Qed.

(* Specifications: *)
(* if a is an element of all A_i, then it is an element of the intersection *)
Lemma in_subsets_then_in_indexed_intersection :
forall a : A,
(forall i : I, in_part a (f i)) -> in_part a indexed_intersection.