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

Lemma bigcup_lift_lift_bigcup {T U} (s1 : set T) (f : T -> set U) :
\bigcup_(x in s1) (lift (f x)) \subset lift (\bigcup_(x in s1) (f x)).