Theorem dfdmf 4797

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 4614	. 2 |- dom A = { w | E. v w A v }
2		nfcv 2308	. . . . 5 |- F/_ y w
3		dfdmf.2	. . . . 5 |- F/_ y A
4		nfcv 2308	. . . . 5 |- F/_ y v
5	2, 3, 4	nfbr 4028	. . . 4 |- F/ y w A v
6		nfv 1516	. . . 4 |- F/ v w A y
7		breq2 3986	. . . 4 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvex 1744	. . 3 |- ( E. v w A v <-> E. y w A y )
9	8	abbii 2282	. 2 |- { w | E. v w A v } = { w | E. y w A y }
10		nfcv 2308	. . . . 5 |- F/_ x w
11		dfdmf.1	. . . . 5 |- F/_ x A
12		nfcv 2308	. . . . 5 |- F/_ x y
13	10, 11, 12	nfbr 4028	. . . 4 |- F/ x w A y
14	13	nfex 1625	. . 3 |- F/ x E. y w A y
15		nfv 1516	. . 3 |- F/ w E. y x A y
16		breq1 3985	. . . 4 |- ( w = x -> ( w A y <-> x A y ) )
17	16	exbidv 1813	. . 3 |- ( w = x -> ( E. y w A y <-> E. y x A y ) )
18	14, 15, 17	cbvab 2290	. 2 |- { w | E. y w A y } = { x | E. y x A y }
19	1, 9, 18	3eqtri 2190	1 |- dom A = { x | E. y x A y }

Syntax hints:   = wceq 1343   E.wex 1480   {cab 2151   F/_wnfc 2295   class class class wbr 3982   dom cdm 4604

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-dm 4614

This theorem is referenced by:  dmopab  4815