Theorem dfdm2 4852

Description: Alternate definition of domain df-dm 4355 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 4520	. . . . . 6 |- `' ( `' A o. A ) = ( `' A o. `' `' A )
2		cocnvcnv2 4832	. . . . . 6 |- ( `' A o. `' `' A ) = ( `' A o. A )
3	1, 2	eqtri 2060	. . . . 5 |- `' ( `' A o. A ) = ( `' A o. A )
4	3	unieqi 3590	. . . 4 |- U. `' ( `' A o. A ) = U. ( `' A o. A )
5	4	unieqi 3590	. . 3 |- U. U. `' ( `' A o. A ) = U. U. ( `' A o. A )
6		unidmrn 4850	. . 3 |- U. U. `' ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
7	5, 6	eqtr3i 2062	. 2 |- U. U. ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
8		df-rn 4356	. . . . 5 |- ran A = dom `' A
9	8	eqcomi 2044	. . . 4 |- dom `' A = ran A
10		dmcoeq 4604	. . . 4 |- ( dom `' A = ran A -> dom ( `' A o. A ) = dom A )
11	9, 10	ax-mp 7	. . 3 |- dom ( `' A o. A ) = dom A
12		rncoeq 4605	. . . . 5 |- ( dom `' A = ran A -> ran ( `' A o. A ) = ran `' A )
13	9, 12	ax-mp 7	. . . 4 |- ran ( `' A o. A ) = ran `' A
14		dfdm4 4527	. . . 4 |- dom A = ran `' A
15	13, 14	eqtr4i 2063	. . 3 |- ran ( `' A o. A ) = dom A
16	11, 15	uneq12i 3095	. 2 |- ( dom ( `' A o. A ) u. ran ( `' A o. A ) ) = ( dom A u. dom A )
17		unidm 3086	. 2 |- ( dom A u. dom A ) = dom A
18	7, 16, 17	3eqtrri 2065	1 |- dom A = U. U. ( `' A o. A )

Syntax hints:   = wceq 1243   u. cun 2915   U.cuni 3580   `'ccnv 4344   dom cdm 4345   ran crn 4346   o. ccom 4349

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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357

This theorem is referenced by: (None)