Theorem dfdm2 5271

Description: Alternate definition of domain df-dm 4735 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 4915	. . . . . 6 |- `' ( `' A o. A ) = ( `' A o. `' `' A )
2		cocnvcnv2 5248	. . . . . 6 |- ( `' A o. `' `' A ) = ( `' A o. A )
3	1, 2	eqtri 2252	. . . . 5 |- `' ( `' A o. A ) = ( `' A o. A )
4	3	unieqi 3903	. . . 4 |- U. `' ( `' A o. A ) = U. ( `' A o. A )
5	4	unieqi 3903	. . 3 |- U. U. `' ( `' A o. A ) = U. U. ( `' A o. A )
6		unidmrn 5269	. . 3 |- U. U. `' ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
7	5, 6	eqtr3i 2254	. 2 |- U. U. ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
8		df-rn 4736	. . . . 5 |- ran A = dom `' A
9	8	eqcomi 2235	. . . 4 |- dom `' A = ran A
10		dmcoeq 5005	. . . 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 5006	. . . . 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 4923	. . . 4 |- dom A = ran `' A
15	13, 14	eqtr4i 2255	. . 3 |- ran ( `' A o. A ) = dom A
16	11, 15	uneq12i 3359	. 2 |- ( dom ( `' A o. A ) u. ran ( `' A o. A ) ) = ( dom A u. dom A )
17		unidm 3350	. 2 |- ( dom A u. dom A ) = dom A
18	7, 16, 17	3eqtrri 2257	1 |- dom A = U. U. ( `' A o. A )

Syntax hints:   = wceq 1397   u. cun 3198   U.cuni 3893   `'ccnv 4724   dom cdm 4725   ran crn 4726   o. ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737

This theorem is referenced by: (None)