Theorem dfrnf 4919

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 4866	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2348	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2348	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4090	. . . 4 |- F/ x v A w
6		nfv 1551	. . . 4 |- F/ v x A w
7		breq1 4047	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1779	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2321	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2348	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2348	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4090	. . . 4 |- F/ y x A w
14	13	nfex 1660	. . 3 |- F/ y E. x x A w
15		nfv 1551	. . 3 |- F/ w E. x x A y
16		breq2 4048	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1848	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2329	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2230	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1373   E.wex 1515   {cab 2191   F/_wnfc 2335   class class class wbr 4044   ran crn 4676

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-cnv 4683  df-dm 4685  df-rn 4686

This theorem is referenced by:  rnopab  4925