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

Lemma eq_bigcup' :
forall (T U : Type) (A B : set T) (F G : T -> set U),
A <--> B ->
(forall x, F x <--> G x) ->
\bigcup_(x in A) F x <--> \bigcup_(x in B) G x.