MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdm2 Unicode version

Theorem dfdm2 5204
Description: Alternate definition of domain df-dm 4699 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 4865 . . . . . 6  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  `' `' A )
2 cocnvcnv2 5184 . . . . . 6  |-  ( `' A  o.  `' `' A )  =  ( `' A  o.  A
)
31, 2eqtri 2303 . . . . 5  |-  `' ( `' A  o.  A
)  =  ( `' A  o.  A )
43unieqi 3837 . . . 4  |-  U. `' ( `' A  o.  A
)  =  U. ( `' A  o.  A
)
54unieqi 3837 . . 3  |-  U. U. `' ( `' A  o.  A )  =  U. U. ( `' A  o.  A )
6 unidmrn 5202 . . 3  |-  U. U. `' ( `' A  o.  A )  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
75, 6eqtr3i 2305 . 2  |-  U. U. ( `' A  o.  A
)  =  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )
8 df-rn 4700 . . . . 5  |-  ran  A  =  dom  `' A
98eqcomi 2287 . . . 4  |-  dom  `' A  =  ran  A
10 dmcoeq 4947 . . . 4  |-  ( dom  `' A  =  ran  A  ->  dom  ( `' A  o.  A )  =  dom  A )
119, 10ax-mp 8 . . 3  |-  dom  ( `' A  o.  A
)  =  dom  A
12 rncoeq 4948 . . . . 5  |-  ( dom  `' A  =  ran  A  ->  ran  ( `' A  o.  A )  =  ran  `' A )
139, 12ax-mp 8 . . . 4  |-  ran  ( `' A  o.  A
)  =  ran  `' A
14 dfdm4 4872 . . . 4  |-  dom  A  =  ran  `' A
1513, 14eqtr4i 2306 . . 3  |-  ran  ( `' A  o.  A
)  =  dom  A
1611, 15uneq12i 3327 . 2  |-  ( dom  ( `' A  o.  A )  u.  ran  ( `' A  o.  A
) )  =  ( dom  A  u.  dom  A )
17 unidm 3318 . 2  |-  ( dom 
A  u.  dom  A
)  =  dom  A
187, 16, 173eqtrri 2308 1  |-  dom  A  =  U. U. ( `' A  o.  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690    o. ccom 4693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701
  Copyright terms: Public domain W3C validator