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Theorem unidmrn 5535
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 5372 . . . 4  |-  Rel  `' A
2 relfld 5531 . . . 4  |-  ( Rel  `' A  ->  U. U. `' A  =  ( dom  `' A  u.  ran  `' A ) )
31, 2ax-mp 5 . . 3  |-  U. U. `' A  =  ( dom  `' A  u.  ran  `' A )
43equncomi 3650 . 2  |-  U. U. `' A  =  ( ran  `' A  u.  dom  `' A )
5 dfdm4 5193 . . 3  |-  dom  A  =  ran  `' A
6 df-rn 5010 . . 3  |-  ran  A  =  dom  `' A
75, 6uneq12i 3656 . 2  |-  ( dom 
A  u.  ran  A
)  =  ( ran  `' A  u.  dom  `' A )
84, 7eqtr4i 2499 1  |-  U. U. `' A  =  ( dom  A  u.  ran  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    u. cun 3474   U.cuni 4245   `'ccnv 4998   dom cdm 4999   ran crn 5000   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  relcnvfld  5536  dfdm2  5537
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