Theorem dfrnf 4965

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 4910	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2372	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2372	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4130	. . . 4 |- F/ x v A w
6		nfv 1574	. . . 4 |- F/ v x A w
7		breq1 4086	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1802	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2345	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2372	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2372	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4130	. . . 4 |- F/ y x A w
14	13	nfex 1683	. . 3 |- F/ y E. x x A w
15		nfv 1574	. . 3 |- F/ w E. x x A y
16		breq2 4087	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1871	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2353	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2254	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1395   E.wex 1538   {cab 2215   F/_wnfc 2359   class class class wbr 4083   ran crn 4720

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  rnopab  4971