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

Lemma bigcup_cons_subset {A B} l (ls : seq A) (f : A -> set B) s :
f l \subset s ->
\bigcup_(x in ls) (f x) \subset s ->
\bigcup_(x in l :: ls) (f x) \subset s.