Theorem dfrnf 5018

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 4963	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2392	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2392	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4172	. . . 4 |- F/ x v A w
6		nfv 1581	. . . 4 |- F/ v x A w
7		breq1 4128	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1809	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2354	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2392	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2392	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4172	. . . 4 |- F/ y x A w
14	13	nfex 1690	. . 3 |- F/ y E. x x A w
15		nfv 1581	. . 3 |- F/ w E. x x A y
16		breq2 4129	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1878	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2364	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2263	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1402   E.wex 1545   {cab 2224   F/_wnfc 2379   class class class wbr 4125   ran crn 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  rnopab  5024