Theorem dfdmf 4776

Description: Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfdmf.1	|- F/_ x A
dfdmf.2	|- F/_ y A

Assertion

Ref	Expression
dfdmf	|- dom A = { x | E. y x A y }

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem dfdmf

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dm 4593	. 2 |- dom A = { w | E. v w A v }
2		nfcv 2299	. . . . 5 |- F/_ y w
3		dfdmf.2	. . . . 5 |- F/_ y A
4		nfcv 2299	. . . . 5 |- F/_ y v
5	2, 3, 4	nfbr 4010	. . . 4 |- F/ y w A v
6		nfv 1508	. . . 4 |- F/ v w A y
7		breq2 3969	. . . 4 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvex 1736	. . 3 |- ( E. v w A v <-> E. y w A y )
9	8	abbii 2273	. 2 |- { w | E. v w A v } = { w | E. y w A y }
10		nfcv 2299	. . . . 5 |- F/_ x w
11		dfdmf.1	. . . . 5 |- F/_ x A
12		nfcv 2299	. . . . 5 |- F/_ x y
13	10, 11, 12	nfbr 4010	. . . 4 |- F/ x w A y
14	13	nfex 1617	. . 3 |- F/ x E. y w A y
15		nfv 1508	. . 3 |- F/ w E. y x A y
16		breq1 3968	. . . 4 |- ( w = x -> ( w A y <-> x A y ) )
17	16	exbidv 1805	. . 3 |- ( w = x -> ( E. y w A y <-> E. y x A y ) )
18	14, 15, 17	cbvab 2281	. 2 |- { w | E. y w A y } = { x | E. y x A y }
19	1, 9, 18	3eqtri 2182	1 |- dom A = { x | E. y x A y }

Syntax hints:   = wceq 1335   E.wex 1472   {cab 2143   F/_wnfc 2286   class class class wbr 3965   dom cdm 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-dm 4593

This theorem is referenced by:  dmopab  4794