Theorem opabrn 29424

Description: Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)

Assertion

Ref	Expression
opabrn	|- ( R = { <. x , y >. | ph } -> ran R = { y | E. x ph } )

Distinct variable group:   x, y, R

Allowed substitution hints:   ph( x, y)

Proof of Theorem opabrn

Step	Hyp	Ref	Expression
1		dfrn2 5311	. 2 |- ran R = { y | E. x x R y }
2		nfopab2 4720	. . . 4 |- F/_ y { <. x , y >. | ph }
3	2	nfeq2 2780	. . 3 |- F/ y R = { <. x , y >. | ph }
4		nfopab1 4719	. . . . 5 |- F/_ x { <. x , y >. | ph }
5	4	nfeq2 2780	. . . 4 |- F/ x R = { <. x , y >. | ph }
6		df-br 4654	. . . . 5 |- ( x R y <-> <. x , y >. e. R )
7		eleq2 2690	. . . . . 6 |- ( R = { <. x , y >. | ph } -> ( <. x , y >. e. R <-> <. x , y >. e. { <. x , y >. | ph } ) )
8		opabid 4982	. . . . . 6 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
9	7, 8	syl6bb 276	. . . . 5 |- ( R = { <. x , y >. | ph } -> ( <. x , y >. e. R <-> ph ) )
10	6, 9	syl5bb 272	. . . 4 |- ( R = { <. x , y >. | ph } -> ( x R y <-> ph ) )
11	5, 10	exbid 2091	. . 3 |- ( R = { <. x , y >. | ph } -> ( E. x x R y <-> E. x ph ) )
12	3, 11	abbid 2740	. 2 |- ( R = { <. x , y >. | ph } -> { y | E. x x R y } = { y | E. x ph } )
13	1, 12	syl5eq 2668	1 |- ( R = { <. x , y >. | ph } -> ran R = { y | E. x ph } )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   <.cop 4183   class class class wbr 4653   {copab 4712   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  fpwrelmapffslem  29507