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Theorem unidmrn 4963
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 4810 . . . 4  |-  Rel  `' A
2 relfld 4959 . . . 4  |-  ( Rel  `' A  ->  U. U. `' A  =  ( dom  `' A  u.  ran  `' A ) )
31, 2ax-mp 7 . . 3  |-  U. U. `' A  =  ( dom  `' A  u.  ran  `' A )
43equncomi 3146 . 2  |-  U. U. `' A  =  ( ran  `' A  u.  dom  `' A )
5 dfdm4 4628 . . 3  |-  dom  A  =  ran  `' A
6 df-rn 4449 . . 3  |-  ran  A  =  dom  `' A
75, 6uneq12i 3152 . 2  |-  ( dom 
A  u.  ran  A
)  =  ( ran  `' A  u.  dom  `' A )
84, 7eqtr4i 2111 1  |-  U. U. `' A  =  ( dom  A  u.  ran  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1289    u. cun 2997   U.cuni 3653   `'ccnv 4437   dom cdm 4438   ran crn 4439   Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  relcnvfld  4964  dfdm2  4965
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