Section BigSetOps.
Variables T I : finType.
Implicit Types (U : {pred T}) (P : pred I) (A B : {set I}) (F :  I -> {set T}).

Lemma bigcup_disjointP U P F  :
reflect (forall i : I, P i -> [disjoint U & F i])
[disjoint U & \bigcup_(i | P i) F i].