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Theorem unidmrn 5218
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 5067 . . . 4  |-  Rel  `' A
2 relfld 5214 . . . 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 3334 . 2  |-  U. U. `' A  =  ( ran  `' A  u.  dom  `' A )
5 dfdm4 4888 . . 3  |-  dom  A  =  ran  `' A
6 df-rn 4716 . . 3  |-  ran  A  =  dom  `' A
75, 6uneq12i 3340 . 2  |-  ( dom 
A  u.  ran  A
)  =  ( ran  `' A  u.  dom  `' A )
84, 7eqtr4i 2319 1  |-  U. U. `' A  =  ( dom  A  u.  ran  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163   U.cuni 3843   `'ccnv 4704   dom cdm 4705   ran crn 4706   Rel wrel 4710
This theorem is referenced by:  relcnvfld  5219  dfdm2  5220  ducidu  25158
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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