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

(* Specifications: *)
Lemma somes_subset {A} (s1 s2 : set (option A)) : s1 \subset s2 -> somes s1 \subset somes s2.
Lemma bigcup_somes {A B} (sA : set A) (s : A -> set (option B)) : somes (\bigcup_(a in sA) s a) <--> \bigcup_(a in sA) somes (s a).
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).