Theorem dfdm2 5317

Description: Alternate definition of domain df-dm 4779 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 4960	. . . . . 6 |- `' ( `' A o. A ) = ( `' A o. `' `' A )
2		cocnvcnv2 5294	. . . . . 6 |- ( `' A o. `' `' A ) = ( `' A o. A )
3	1, 2	eqtri 2259	. . . . 5 |- `' ( `' A o. A ) = ( `' A o. A )
4	3	unieqi 3940	. . . 4 |- U. `' ( `' A o. A ) = U. ( `' A o. A )
5	4	unieqi 3940	. . 3 |- U. U. `' ( `' A o. A ) = U. U. ( `' A o. A )
6		unidmrn 5315	. . 3 |- U. U. `' ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
7	5, 6	eqtr3i 2261	. 2 |- U. U. ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
8		df-rn 4780	. . . . 5 |- ran A = dom `' A
9	8	eqcomi 2242	. . . 4 |- dom `' A = ran A
10		dmcoeq 5050	. . . 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 5051	. . . . 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 4968	. . . 4 |- dom A = ran `' A
15	13, 14	eqtr4i 2262	. . 3 |- ran ( `' A o. A ) = dom A
16	11, 15	uneq12i 3381	. 2 |- ( dom ( `' A o. A ) u. ran ( `' A o. A ) ) = ( dom A u. dom A )
17		unidm 3372	. 2 |- ( dom A u. dom A ) = dom A
18	7, 16, 17	3eqtrri 2264	1 |- dom A = U. U. ( `' A o. A )

Syntax hints:   = wceq 1402   u. cun 3218   U.cuni 3930   `'ccnv 4768   dom cdm 4769   ran crn 4770   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781

This theorem is referenced by: (None)