Instance eq_bigcap T U : Proper (set_eq ==> pointwise_relation T (@set_eq U) ==> set_eq) bigcap.

Lemma setI_bigcup_assoc {A B} (s1 : set B) (s2 : set A) (s3 : A -> set B) :
s1 :&: (\bigcup_(x in s2) s3 x) <--> \bigcup_(x in s2) (s1 :&: (s3 x)).