Definition somes {A} (s : set (option A)) : set A :=
  [set x | Some x \in s].

Lemma bigcup_setI {T U} (s1 : set T) (s2 : set U) F : \bigcup_(x in s1) (s2 :&: F x) <--> s2 :&: \bigcup_(x in s1) (F x).