(** Lemmas about [bigcap] *)

(* Specifications: *)
Lemma bigcap_set0 (T U : Type) (F : T -> set U) :
\bigcap_(x in set0) F x <--> setT.
Lemma incl_bigcapl T U A B (F : T -> set U) : B \subset A ->
\bigcap_(x in A) F x \subset \bigcap_(x in B) F x.
Lemma eq_bigcapr (T U : Type) (A : set T) (F G : T -> set U) :
(forall x : T, F x <--> G x) ->
\bigcap_(x in A) F x <--> \bigcap_(x in A) G x.
Lemma eq_bigcapl T U A B (F : T -> set U) : A <--> B ->
\bigcap_(x in A) F x <--> \bigcap_(x in B) F x.