Theorem dfrnf 4575

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	|- F/_ x A
dfrnf.2	|- F/_ y A

Assertion

Ref	Expression
dfrnf	|- ran A = { y | E. x x A y }

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem dfrnf

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 4523	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2178	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2178	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 3808	. . . 4 |- F/ x v A w
6		nfv 1421	. . . 4 |- F/ v x A w
7		breq1 3767	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1639	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2153	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2178	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2178	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 3808	. . . 4 |- F/ y x A w
14	13	nfex 1528	. . 3 |- F/ y E. x x A w
15		nfv 1421	. . 3 |- F/ w E. x x A y
16		breq2 3768	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1706	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2160	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2064	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1243   E.wex 1381   {cab 2026   F/_wnfc 2165   class class class wbr 3764   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  rnopab  4581