Theorem dfrnf 4593

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 4541	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2219	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2219	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 3829	. . . 4 |- F/ x v A w
6		nfv 1461	. . . 4 |- F/ v x A w
7		breq1 3788	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1679	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2194	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2219	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2219	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 3829	. . . 4 |- F/ y x A w
14	13	nfex 1568	. . 3 |- F/ y E. x x A w
15		nfv 1461	. . 3 |- F/ w E. x x A y
16		breq2 3789	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1746	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2201	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2105	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1284   E.wex 1421   {cab 2067   F/_wnfc 2206   class class class wbr 3785   ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  rnopab  4599