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

Lemma bigcup_setI_cons_subset_r :
forall (A B : Type) (l : A) (ls : seq A) (f : A -> set B) (s1 s2 : set B) (s3 : set A),
s3 l ->
s1 \subset f l ->
s2 \subset \bigcup_(x in ls :&: s3) f x ->
s1 :|: s2 \subset \bigcup_(x in (l :: ls) :&: s3) f x.