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Theorem dfdm2 4830
Description: Alternate definition of domain df-dm 4333 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
StepHypRef Expression
1 cnvco 4498 . . . . . 6  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  `' `' A )
2 cocnvcnv2 4810 . . . . . 6  |-  ( `' A  o.  `' `' A )  =  ( `' A  o.  A
)
31, 2eqtri 2060 . . . . 5  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  A )
43unieqi 3587 . . . 4  |-  U. `' ( `' A  o.  A
)  =  U. ( `' A  o.  A
)
54unieqi 3587 . . 3  |-  U. U. `' ( `' A  o.  A )  =  U. U. ( `' A  o.  A )
6 unidmrn 4828 . . 3  |-  U. U. `' ( `' A  o.  A )  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
75, 6eqtr3i 2062 . 2  |-  U. U. ( `' A  o.  A
)  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
8 df-rn 4334 . . . . 5  |-  ran  A  =  dom  `' A
98eqcomi 2044 . . . 4  |-  dom  `' A  =  ran  A
10 dmcoeq 4582 . . . 4  |-  ( dom  `' A  =  ran  A  ->  dom  ( `' A  o.  A )  =  dom  A )
119, 10ax-mp 7 . . 3  |-  dom  ( `' A  o.  A
)  =  dom  A
12 rncoeq 4583 . . . . 5  |-  ( dom  `' A  =  ran  A  ->  ran  ( `' A  o.  A )  =  ran  `' A )
139, 12ax-mp 7 . . . 4  |-  ran  ( `' A  o.  A
)  =  ran  `' A
14 dfdm4 4505 . . . 4  |-  dom  A  =  ran  `' A
1513, 14eqtr4i 2063 . . 3  |-  ran  ( `' A  o.  A
)  =  dom  A
1611, 15uneq12i 3092 . 2  |-  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )  =  ( dom  A  u.  dom  A )
17 unidm 3083 . 2  |-  ( dom 
A  u.  dom  A
)  =  dom  A
187, 16, 173eqtrri 2065 1  |-  dom  A  =  U. U. ( `' A  o.  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1243    u. cun 2912   U.cuni 3577   `'ccnv 4322   dom cdm 4323   ran crn 4324    o. ccom 4327
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 3872  ax-pow 3924  ax-pr 3941
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 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-xp 4329  df-rel 4330  df-cnv 4331  df-co 4332  df-dm 4333  df-rn 4334  df-res 4335
This theorem is referenced by: (None)
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