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

Lemma bigcap_setU_distr:
forall (U T : Type) (s1 s2 : set U) (f : U -> set T),
\bigcap_(i in s1) f i :&: \bigcap_(i in s2) f i <--> \bigcap_(i in s1 :|: s2) f i.