Theorem dfdmf 4818

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 4635	. 2 |- dom A = { w | E. v w A v }
2		nfcv 2319	. . . . 5 |- F/_ y w
3		dfdmf.2	. . . . 5 |- F/_ y A
4		nfcv 2319	. . . . 5 |- F/_ y v
5	2, 3, 4	nfbr 4048	. . . 4 |- F/ y w A v
6		nfv 1528	. . . 4 |- F/ v w A y
7		breq2 4006	. . . 4 |- ( v = y -> ( w A v <-> w A y ) )
8	5, 6, 7	cbvex 1756	. . 3 |- ( E. v w A v <-> E. y w A y )
9	8	abbii 2293	. 2 |- { w | E. v w A v } = { w | E. y w A y }
10		nfcv 2319	. . . . 5 |- F/_ x w
11		dfdmf.1	. . . . 5 |- F/_ x A
12		nfcv 2319	. . . . 5 |- F/_ x y
13	10, 11, 12	nfbr 4048	. . . 4 |- F/ x w A y
14	13	nfex 1637	. . 3 |- F/ x E. y w A y
15		nfv 1528	. . 3 |- F/ w E. y x A y
16		breq1 4005	. . . 4 |- ( w = x -> ( w A y <-> x A y ) )
17	16	exbidv 1825	. . 3 |- ( w = x -> ( E. y w A y <-> E. y x A y ) )
18	14, 15, 17	cbvab 2301	. 2 |- { w | E. y w A y } = { x | E. y x A y }
19	1, 9, 18	3eqtri 2202	1 |- dom A = { x | E. y x A y }

Syntax hints:   = wceq 1353   E.wex 1492   {cab 2163   F/_wnfc 2306   class class class wbr 4002   dom cdm 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-dm 4635

This theorem is referenced by:  dmopab  4836