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

Lemma incl_bigcup_compat :
forall (T U : Type) (A B : set T) (F G : T -> set U),
A \subset B ->
(forall x : T, F x \subset G x) ->
\bigcup_(x in A) F x \subset \bigcup_(x in B) G x.