intros.
unfold indexed_intersection in |- *.
simpl in |- *.
assumption.
Qed.

(* contrarily, if for some i \in I a is not in A_i, then neither is it in *)
(* the indexed intersection *)
Lemma not_in_part_then_not_in_indexed_intersection :
forall a : A,
(exists i : I, ~ in_part a (f i)) -> ~ in_part a indexed_intersection.