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Theorem dfdm2 5522
Description: Alternate definition of domain df-dm 4998 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 5177 . . . . . 6  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  `' `' A )
2 cocnvcnv2 5502 . . . . . 6  |-  ( `' A  o.  `' `' A )  =  ( `' A  o.  A
)
31, 2eqtri 2483 . . . . 5  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  A )
43unieqi 4244 . . . 4  |-  U. `' ( `' A  o.  A
)  =  U. ( `' A  o.  A
)
54unieqi 4244 . . 3  |-  U. U. `' ( `' A  o.  A )  =  U. U. ( `' A  o.  A )
6 unidmrn 5520 . . 3  |-  U. U. `' ( `' A  o.  A )  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
75, 6eqtr3i 2485 . 2  |-  U. U. ( `' A  o.  A
)  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
8 df-rn 4999 . . . . 5  |-  ran  A  =  dom  `' A
98eqcomi 2467 . . . 4  |-  dom  `' A  =  ran  A
10 dmcoeq 5254 . . . 4  |-  ( dom  `' A  =  ran  A  ->  dom  ( `' A  o.  A )  =  dom  A )
119, 10ax-mp 5 . . 3  |-  dom  ( `' A  o.  A
)  =  dom  A
12 rncoeq 5255 . . . . 5  |-  ( dom  `' A  =  ran  A  ->  ran  ( `' A  o.  A )  =  ran  `' A )
139, 12ax-mp 5 . . . 4  |-  ran  ( `' A  o.  A
)  =  ran  `' A
14 dfdm4 5184 . . . 4  |-  dom  A  =  ran  `' A
1513, 14eqtr4i 2486 . . 3  |-  ran  ( `' A  o.  A
)  =  dom  A
1611, 15uneq12i 3642 . 2  |-  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )  =  ( dom  A  u.  dom  A )
17 unidm 3633 . 2  |-  ( dom 
A  u.  dom  A
)  =  dom  A
187, 16, 173eqtrri 2488 1  |-  dom  A  =  U. U. ( `' A  o.  A )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    u. cun 3459   U.cuni 4235   `'ccnv 4987   dom cdm 4988   ran crn 4989    o. ccom 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000
This theorem is referenced by: (None)
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