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

Lemma imset_union_incl {U T : Type} (s1 s2 : set U) (f : U -> T) :
f @: (s1 :|: s2) \subset (f @: s1) :|: (f @: s2).