Theorem dfrnf 5364

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 5311	. 2 |- ran A = { w | E. v v A w }
2		nfcv 2764	. . . . 5 |- F/_ x v
3		dfrnf.1	. . . . 5 |- F/_ x A
4		nfcv 2764	. . . . 5 |- F/_ x w
5	2, 3, 4	nfbr 4699	. . . 4 |- F/ x v A w
6		nfv 1843	. . . 4 |- F/ v x A w
7		breq1 4656	. . . 4 |- ( v = x -> ( v A w <-> x A w ) )
8	5, 6, 7	cbvex 2272	. . 3 |- ( E. v v A w <-> E. x x A w )
9	8	abbii 2739	. 2 |- { w | E. v v A w } = { w | E. x x A w }
10		nfcv 2764	. . . . 5 |- F/_ y x
11		dfrnf.2	. . . . 5 |- F/_ y A
12		nfcv 2764	. . . . 5 |- F/_ y w
13	10, 11, 12	nfbr 4699	. . . 4 |- F/ y x A w
14	13	nfex 2154	. . 3 |- F/ y E. x x A w
15		nfv 1843	. . 3 |- F/ w E. x x A y
16		breq2 4657	. . . 4 |- ( w = y -> ( x A w <-> x A y ) )
17	16	exbidv 1850	. . 3 |- ( w = y -> ( E. x x A w <-> E. x x A y ) )
18	14, 15, 17	cbvab 2746	. 2 |- { w | E. x x A w } = { y | E. x x A y }
19	1, 9, 18	3eqtri 2648	1 |- ran A = { y | E. x x A y }

Syntax hints:   = wceq 1483   E.wex 1704   {cab 2608   F/_wnfc 2751   class class class wbr 4653   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  rnopab  5370