Theorem dfrnf 4708

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 4655	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2235	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2235	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 3911	. . . 4 |- F/ x v A w
6		nfv 1473	. . . 4 |- F/ v x A w
7		breq1 3870	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1693	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2210	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2235	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2235	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 3911	. . . 4 |- F/ y x A w
14	13	nfex 1580	. . 3 |- F/ y E. x x A w
15		nfv 1473	. . 3 |- F/ w E. x x A y
16		breq2 3871	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1760	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2217	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2119	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1296   E.wex 1433   {cab 2081   F/_wnfc 2222   class class class wbr 3867   ran crn 4468

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-opab 3922  df-cnv 4475  df-dm 4477  df-rn 4478

This theorem is referenced by:  rnopab  4714