Theorem dfdmf 4528

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 4355	. 2 |- dom A = { w | E. v w A v }
2		nfcv 2178	. . . . 5 |- F/_ y w
3		dfdmf.2	. . . . 5 |- F/_ y A
4		nfcv 2178	. . . . 5 |- F/_ y v
5	2, 3, 4	nfbr 3808	. . . 4 |- F/ y w A v
6		nfv 1421	. . . 4 |- F/ v w A y
7		breq2 3768	. . . 4 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvex 1639	. . 3 |- ( E. v w A v <-> E. y w A y )
9	8	abbii 2153	. 2 |- { w | E. v w A v } = { w | E. y w A y }
10		nfcv 2178	. . . . 5 |- F/_ x w
11		dfdmf.1	. . . . 5 |- F/_ x A
12		nfcv 2178	. . . . 5 |- F/_ x y
13	10, 11, 12	nfbr 3808	. . . 4 |- F/ x w A y
14	13	nfex 1528	. . 3 |- F/ x E. y w A y
15		nfv 1421	. . 3 |- F/ w E. y x A y
16		breq1 3767	. . . 4 |- ( w = x -> ( w A y <-> x A y ) )
17	16	exbidv 1706	. . 3 |- ( w = x -> ( E. y w A y <-> E. y x A y ) )
18	14, 15, 17	cbvab 2160	. 2 |- { w | E. y w A y } = { x | E. y x A y }
19	1, 9, 18	3eqtri 2064	1 |- dom A = { x | E. y x A y }

Syntax hints:   = wceq 1243   E.wex 1381   {cab 2026   F/_wnfc 2165   class class class wbr 3764   dom cdm 4345

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by:  dmopab  4546