Theorem dfrnf 4861

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 4808	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2317	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2317	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4044	. . . 4 |- F/ x v A w
6		nfv 1526	. . . 4 |- F/ v x A w
7		breq1 4001	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1754	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2291	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2317	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2317	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4044	. . . 4 |- F/ y x A w
14	13	nfex 1635	. . 3 |- F/ y E. x x A w
15		nfv 1526	. . 3 |- F/ w E. x x A y
16		breq2 4002	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1823	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2299	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2200	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1353   E.wex 1490   {cab 2161   F/_wnfc 2304   class class class wbr 3998   ran crn 4621

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by:  rnopab  4867