Theorem dfdm2 5145

Description: Alternate definition of domain df-dm 4621 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 4796	. . . . . 6 |- `' ( `' A o. A ) = ( `' A o. `' `' A )
2		cocnvcnv2 5122	. . . . . 6 |- ( `' A o. `' `' A ) = ( `' A o. A )
3	1, 2	eqtri 2191	. . . . 5 |- `' ( `' A o. A ) = ( `' A o. A )
4	3	unieqi 3806	. . . 4 |- U. `' ( `' A o. A ) = U. ( `' A o. A )
5	4	unieqi 3806	. . 3 |- U. U. `' ( `' A o. A ) = U. U. ( `' A o. A )
6		unidmrn 5143	. . 3 |- U. U. `' ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
7	5, 6	eqtr3i 2193	. 2 |- U. U. ( `' A o. A ) = ( dom ( `' A o. A ) u. ran ( `' A o. A ) )
8		df-rn 4622	. . . . 5 |- ran A = dom `' A
9	8	eqcomi 2174	. . . 4 |- dom `' A = ran A
10		dmcoeq 4883	. . . 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 4884	. . . . 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 4803	. . . 4 |- dom A = ran `' A
15	13, 14	eqtr4i 2194	. . 3 |- ran ( `' A o. A ) = dom A
16	11, 15	uneq12i 3279	. 2 |- ( dom ( `' A o. A ) u. ran ( `' A o. A ) ) = ( dom A u. dom A )
17		unidm 3270	. 2 |- ( dom A u. dom A ) = dom A
18	7, 16, 17	3eqtrri 2196	1 |- dom A = U. U. ( `' A o. A )

Syntax hints:   = wceq 1348   u. cun 3119   U.cuni 3796   `'ccnv 4610   dom cdm 4611   ran crn 4612   o. ccom 4615

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623

This theorem is referenced by: (None)