Coercion seq_In T : seq T -> set T := fun s x => List.In x s.Coercion list_In T : list T -> set T := fun s x => List.In x s.

Lemma setI_bigcup_assoc {A B} (s1 : set B) (s2 : set A) (s3 : A -> set B) : s1 :&: (\bigcup_(x in s2) s3 x) <--> \bigcup_(x in s2) (s1 :&: (s3 x)).