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 bigcup_lift_lift_bigcup {T U} (s1 : set T) (f : T -> set U) : \bigcup_(x in s1) (lift (f x)) \subset lift (\bigcup_(x in s1) (f x)).