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Theorem dfdm2 5478
Description: Alternate definition of domain df-dm 4959 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 5134 . . . . . 6  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  `' `' A )
2 cocnvcnv2 5458 . . . . . 6  |-  ( `' A  o.  `' `' A )  =  ( `' A  o.  A
)
31, 2eqtri 2483 . . . . 5  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  A )
43unieqi 4209 . . . 4  |-  U. `' ( `' A  o.  A
)  =  U. ( `' A  o.  A
)
54unieqi 4209 . . 3  |-  U. U. `' ( `' A  o.  A )  =  U. U. ( `' A  o.  A )
6 unidmrn 5476 . . 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 4960 . . . . 5  |-  ran  A  =  dom  `' A
98eqcomi 2467 . . . 4  |-  dom  `' A  =  ran  A
10 dmcoeq 5211 . . . 4  |-  ( dom  `' A  =  ran  A  ->  dom  ( `' A  o.  A )  =  dom  A )
119, 10ax-mp 5 . . 3  |-  dom  ( `' A  o.  A
)  =  dom  A
12 rncoeq 5212 . . . . 5  |-  ( dom  `' A  =  ran  A  ->  ran  ( `' A  o.  A )  =  ran  `' A )
139, 12ax-mp 5 . . . 4  |-  ran  ( `' A  o.  A
)  =  ran  `' A
14 dfdm4 5141 . . . 4  |-  dom  A  =  ran  `' A
1513, 14eqtr4i 2486 . . 3  |-  ran  ( `' A  o.  A
)  =  dom  A
1611, 15uneq12i 3617 . 2  |-  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )  =  ( dom  A  u.  dom  A )
17 unidm 3608 . 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 1370    u. cun 3435   U.cuni 4200   `'ccnv 4948   dom cdm 4949   ran crn 4950    o. ccom 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961
This theorem is referenced by: (None)
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