Theorem dfrnf 4938

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 4884	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2350	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2350	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4106	. . . 4 |- F/ x v A w
6		nfv 1552	. . . 4 |- F/ v x A w
7		breq1 4062	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1780	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2323	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2350	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2350	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4106	. . . 4 |- F/ y x A w
14	13	nfex 1661	. . 3 |- F/ y E. x x A w
15		nfv 1552	. . 3 |- F/ w E. x x A y
16		breq2 4063	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1849	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2331	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2232	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1373   E.wex 1516   {cab 2193   F/_wnfc 2337   class class class wbr 4059   ran crn 4694

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-cnv 4701  df-dm 4703  df-rn 4704

This theorem is referenced by:  rnopab  4944