Theorem rnoprab2

Description: The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012)

	Ref	Expression
Assertion	rnoprab2	|- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x e. A E. y e. B ph }

Proof

Step	Hyp	Ref	Expression
1		rnoprab	|- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) }
2		r2ex	|- ( E. x e. A E. y e. B ph <-> E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) )
3	2	abbii	|- { z | E. x e. A E. y e. B ph } = { z | E. x E. y ( ( x e. A /\ y e. B ) /\ ph ) }
4	1 3	eqtr4i	|- ran { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) } = { z | E. x e. A E. y e. B ph }