Theorem xrninxp2

Description: Intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 8-Apr-2020)

	Ref	Expression
Assertion	xrninxp2	|- ( ( R |X. S ) i^i ( A X. ( B X. C ) ) ) = { <. u , x >. | ( x e. ( B X. C ) /\ ( u e. A /\ u ( R |X. S ) x ) ) }

Proof

Step	Hyp	Ref	Expression
1		inxp2	|- ( ( R |X. S ) i^i ( A X. ( B X. C ) ) ) = { <. u , x >. | ( ( u e. A /\ x e. ( B X. C ) ) /\ u ( R |X. S ) x ) }
2		an21	|- ( ( ( u e. A /\ x e. ( B X. C ) ) /\ u ( R |X. S ) x ) <-> ( x e. ( B X. C ) /\ ( u e. A /\ u ( R |X. S ) x ) ) )
3	2	opabbii	|- { <. u , x >. | ( ( u e. A /\ x e. ( B X. C ) ) /\ u ( R |X. S ) x ) } = { <. u , x >. | ( x e. ( B X. C ) /\ ( u e. A /\ u ( R |X. S ) x ) ) }
4	1 3	eqtri	|- ( ( R |X. S ) i^i ( A X. ( B X. C ) ) ) = { <. u , x >. | ( x e. ( B X. C ) /\ ( u e. A /\ u ( R |X. S ) x ) ) }