(* Picks A_i for i\in I *)
 
(** - so, given an index set $I$ and an indexing function $f$ mapping 
 this set to the part_set (powerset) of X,  (ie. $f(i) = A_i\ (\in (part\_set A))$,) 
 then (indexed_intersection f) will be $\bigcap_{i\in I} A_i$. *)

Definition indexed_intersection : part_set A.

Lemma indexed_intersection_indep_of_indexing : forall (f : MAP I (part_set A)) (g : MAP J (part_set A)), Equal (range f) (range g) -> Equal (indexed_intersection f) (indexed_intersection g).