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

Lemma isSome_subset {A : Type} (s1 s2 s1' s2' : set (option A)) :
isSome :&: s1 \subset isSome :&: s2 ->
isSome :&: (s1 :|: ([set None] :&: s1')) \subset isSome :&: (s2 :|: ([set None] :&: s2')).