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

Lemma imset_bigcup_incl_l :
forall {T U V : Type} (f : U -> V) (A : set T) (F : T -> set U),
f @: (\bigcup_(x in A) F x) \subset \bigcup_(x in A) f @: F x.