Theorem dfrnf 4845

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 4792	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2308	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2308	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4028	. . . 4 |- F/ x v A w
6		nfv 1516	. . . 4 |- F/ v x A w
7		breq1 3985	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1744	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2282	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2308	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2308	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4028	. . . 4 |- F/ y x A w
14	13	nfex 1625	. . 3 |- F/ y E. x x A w
15		nfv 1516	. . 3 |- F/ w E. x x A y
16		breq2 3986	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1813	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2290	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2190	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1343   E.wex 1480   {cab 2151   F/_wnfc 2295   class class class wbr 3982   ran crn 4605

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  rnopab  4851