Theorem dfrnf 4979

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 4924	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2375	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2375	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4140	. . . 4 |- F/ x v A w
6		nfv 1577	. . . 4 |- F/ v x A w
7		breq1 4096	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1804	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2347	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2375	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2375	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4140	. . . 4 |- F/ y x A w
14	13	nfex 1686	. . 3 |- F/ y E. x x A w
15		nfv 1577	. . 3 |- F/ w E. x x A y
16		breq2 4097	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1873	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2356	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2256	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1398   E.wex 1541   {cab 2217   F/_wnfc 2362   class class class wbr 4093   ran crn 4732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  rnopab  4985