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

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