Theorem dfrnf 4788

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 4735	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2282	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2282	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 3982	. . . 4 |- F/ x v A w
6		nfv 1509	. . . 4 |- F/ v x A w
7		breq1 3940	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1730	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2256	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2282	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2282	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 3982	. . . 4 |- F/ y x A w
14	13	nfex 1617	. . 3 |- F/ y E. x x A w
15		nfv 1509	. . 3 |- F/ w E. x x A y
16		breq2 3941	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1798	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2264	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2165	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1332   E.wex 1469   {cab 2126   F/_wnfc 2269   class class class wbr 3937   ran crn 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  rnopab  4794