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Theorem dfdm2 5393
Description: Alternate definition of domain df-dm 4880 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 5048 . . . . . 6  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  `' `' A )
2 cocnvcnv2 5373 . . . . . 6  |-  ( `' A  o.  `' `' A )  =  ( `' A  o.  A
)
31, 2eqtri 2455 . . . . 5  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  A )
43unieqi 4017 . . . 4  |-  U. `' ( `' A  o.  A
)  =  U. ( `' A  o.  A
)
54unieqi 4017 . . 3  |-  U. U. `' ( `' A  o.  A )  =  U. U. ( `' A  o.  A )
6 unidmrn 5391 . . 3  |-  U. U. `' ( `' A  o.  A )  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
75, 6eqtr3i 2457 . 2  |-  U. U. ( `' A  o.  A
)  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
8 df-rn 4881 . . . . 5  |-  ran  A  =  dom  `' A
98eqcomi 2439 . . . 4  |-  dom  `' A  =  ran  A
10 dmcoeq 5130 . . . 4  |-  ( dom  `' A  =  ran  A  ->  dom  ( `' A  o.  A )  =  dom  A )
119, 10ax-mp 8 . . 3  |-  dom  ( `' A  o.  A
)  =  dom  A
12 rncoeq 5131 . . . . 5  |-  ( dom  `' A  =  ran  A  ->  ran  ( `' A  o.  A )  =  ran  `' A )
139, 12ax-mp 8 . . . 4  |-  ran  ( `' A  o.  A
)  =  ran  `' A
14 dfdm4 5055 . . . 4  |-  dom  A  =  ran  `' A
1513, 14eqtr4i 2458 . . 3  |-  ran  ( `' A  o.  A
)  =  dom  A
1611, 15uneq12i 3491 . 2  |-  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )  =  ( dom  A  u.  dom  A )
17 unidm 3482 . 2  |-  ( dom 
A  u.  dom  A
)  =  dom  A
187, 16, 173eqtrri 2460 1  |-  dom  A  =  U. U. ( `' A  o.  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3310   U.cuni 4007   `'ccnv 4869   dom cdm 4870   ran crn 4871    o. ccom 4874
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882
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