Theorem dfrnf 4886

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 4833	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2332	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2332	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4064	. . . 4 |- F/ x v A w
6		nfv 1539	. . . 4 |- F/ v x A w
7		breq1 4021	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1767	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2305	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2332	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2332	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4064	. . . 4 |- F/ y x A w
14	13	nfex 1648	. . 3 |- F/ y E. x x A w
15		nfv 1539	. . 3 |- F/ w E. x x A y
16		breq2 4022	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1836	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2313	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2214	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1364   E.wex 1503   {cab 2175   F/_wnfc 2319   class class class wbr 4018   ran crn 4645

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4652  df-dm 4654  df-rn 4655

This theorem is referenced by:  rnopab  4892