Theorem dfdm2 5138

Description: Alternate definition of domain df-dm 4614 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.)

Assertion

Ref	Expression
dfdm2	|- dom A = U. U. ( `' A o. A )

Proof of Theorem dfdm2

Step	Hyp	Ref	Expression
1		cnvco 4789	. . . . . 6 |- `' ( `' A o. A ) = ( `' A o. `' `' A )
2		cocnvcnv2 5115	. . . . . 6 |- ( `' A o. `' `' A ) = ( `' A o. A )
3	1, 2	eqtri 2186	. . . . 5 |- `' ( `' A o. A ) = ( `' A o. A )
4	3	unieqi 3799	. . . 4 |- U. `' ( `' A o. A ) = U. ( `' A o. A )
5	4	unieqi 3799	. . 3 |- U. U. `' ( `' A o. A ) = U. U. ( `' A o. A )
6		unidmrn 5136	. . 3 |- U. U. `' ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
7	5, 6	eqtr3i 2188	. 2 |- U. U. ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
8		df-rn 4615	. . . . 5 |- ran A = dom `' A
9	8	eqcomi 2169	. . . 4 |- dom `' A = ran A
10		dmcoeq 4876	. . . 4 |- ( dom `' A = ran A -> dom ( `' A o. A ) = dom A )
11	9, 10	ax-mp 5	. . 3 |- dom ( `' A o. A ) = dom A
12		rncoeq 4877	. . . . 5 |- ( dom `' A = ran A -> ran ( `' A o. A ) = ran `' A )
13	9, 12	ax-mp 5	. . . 4 |- ran ( `' A o. A ) = ran `' A
14		dfdm4 4796	. . . 4 |- dom A = ran `' A
15	13, 14	eqtr4i 2189	. . 3 |- ran ( `' A o. A ) = dom A
16	11, 15	uneq12i 3274	. 2 |- ( dom ( `' A o. A ) u. ran ( `' A o. A ) ) = ( dom A u. dom A )
17		unidm 3265	. 2 |- ( dom A u. dom A ) = dom A
18	7, 16, 17	3eqtrri 2191	1 |- dom A = U. U. ( `' A o. A )

Syntax hints:   = wceq 1343   u. cun 3114   U.cuni 3789   `'ccnv 4603   dom cdm 4604   ran crn 4605   o. ccom 4608

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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616

This theorem is referenced by: (None)