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Theorem unidmrn 5202
Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.)
Assertion
Ref Expression
unidmrn  |-  U. U. `' A  =  ( dom  A  u.  ran  A
)

Proof of Theorem unidmrn
StepHypRef Expression
1 relcnv 5051 . . . 4  |-  Rel  `' A
2 relfld 5198 . . . 4  |-  ( Rel  `' A  ->  U. U. `' A  =  ( dom  `' A  u.  ran  `' A ) )
31, 2ax-mp 8 . . 3  |-  U. U. `' A  =  ( dom  `' A  u.  ran  `' A )
43equncomi 3321 . 2  |-  U. U. `' A  =  ( ran  `' A  u.  dom  `' A )
5 dfdm4 4872 . . 3  |-  dom  A  =  ran  `' A
6 df-rn 4700 . . 3  |-  ran  A  =  dom  `' A
75, 6uneq12i 3327 . 2  |-  ( dom 
A  u.  ran  A
)  =  ( ran  `' A  u.  dom  `' A )
84, 7eqtr4i 2306 1  |-  U. U. `' A  =  ( dom  A  u.  ran  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   Rel wrel 4694
This theorem is referenced by:  relcnvfld  5203  dfdm2  5204  ducidu  25055
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-dm 4699  df-rn 4700
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