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

Lemma imset_union_set_eq:
forall (U T : Type) (s1 s2 : set U) (f : U -> T),
f @: (s1 :|: s2) <--> f @: s1 :|: f @: s2.