Theorem rnopab

Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995) (Revised by Mario Carneiro, 4-Dec-2016)

	Ref	Expression
Assertion	rnopab	|- ran { <. x , y >. | ph } = { y | E. x ph }

Proof

Step	Hyp	Ref	Expression
1		nfopab1	|- F/_ x { <. x , y >. | ph }
2		nfopab2	|- F/_ y { <. x , y >. | ph }
3	1 2	dfrnf	|- ran { <. x , y >. | ph } = { y | E. x x { <. x , y >. | ph } y }
4		df-br	|- ( x { <. x , y >. | ph } y <-> <. x , y >. e. { <. x , y >. | ph } )
5		opabidw	|- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
6	4 5	bitri	|- ( x { <. x , y >. | ph } y <-> ph )
7	6	exbii	|- ( E. x x { <. x , y >. | ph } y <-> E. x ph )
8	7	abbii	|- { y | E. x x { <. x , y >. | ph } y } = { y | E. x ph }
9	3 8	eqtri	|- ran { <. x , y >. | ph } = { y | E. x ph }