Theorem dfrnf 4907

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 4854	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2339	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2339	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4079	. . . 4 |- F/ x v A w
6		nfv 1542	. . . 4 |- F/ v x A w
7		breq1 4036	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 1770	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2312	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2339	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2339	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4079	. . . 4 |- F/ y x A w
14	13	nfex 1651	. . 3 |- F/ y E. x x A w
15		nfv 1542	. . 3 |- F/ w E. x x A y
16		breq2 4037	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1839	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2320	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2221	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1364   E.wex 1506   {cab 2182   F/_wnfc 2326   class class class wbr 4033   ran crn 4664

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  rnopab  4913