Theorem dfdmf 4546

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 4373	. 2 |- dom A = { w | E. v w A v }
2		nfcv 2219	. . . . 5 |- F/_ y w
3		dfdmf.2	. . . . 5 |- F/_ y A
4		nfcv 2219	. . . . 5 |- F/_ y v
5	2, 3, 4	nfbr 3829	. . . 4 |- F/ y w A v
6		nfv 1461	. . . 4 |- F/ v w A y
7		breq2 3789	. . . 4 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvex 1679	. . 3 |- ( E. v w A v <-> E. y w A y )
9	8	abbii 2194	. 2 |- { w | E. v w A v } = { w | E. y w A y }
10		nfcv 2219	. . . . 5 |- F/_ x w
11		dfdmf.1	. . . . 5 |- F/_ x A
12		nfcv 2219	. . . . 5 |- F/_ x y
13	10, 11, 12	nfbr 3829	. . . 4 |- F/ x w A y
14	13	nfex 1568	. . 3 |- F/ x E. y w A y
15		nfv 1461	. . 3 |- F/ w E. y x A y
16		breq1 3788	. . . 4 |- ( w = x -> ( w A y <-> x A y ) )
17	16	exbidv 1746	. . 3 |- ( w = x -> ( E. y w A y <-> E. y x A y ) )
18	14, 15, 17	cbvab 2201	. 2 |- { w | E. y w A y } = { x | E. y x A y }
19	1, 9, 18	3eqtri 2105	1 |- dom A = { x | E. y x A y }

Syntax hints:   = wceq 1284   E.wex 1421   {cab 2067   F/_wnfc 2206   class class class wbr 3785   dom cdm 4363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-dm 4373

This theorem is referenced by:  dmopab  4564