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Theorem domfldref 25061
Description: The domain of a reflexive relation is equal to its field . (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
domfldref  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  U. U. R )
Distinct variable group:    x, R

Proof of Theorem domfldref
StepHypRef Expression
1 relfld 5198 . . 3  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
2 domrngref 25060 . . . 4  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  ran  R
)
3 uneq2 3323 . . . . . 6  |-  ( dom 
R  =  ran  R  ->  ( dom  R  u.  dom  R )  =  ( dom  R  u.  ran  R ) )
43eqcomd 2288 . . . . 5  |-  ( dom 
R  =  ran  R  ->  ( dom  R  u.  ran  R )  =  ( dom  R  u.  dom  R ) )
5 eqtr 2300 . . . . . . 7  |-  ( ( U. U. R  =  ( dom  R  u.  ran  R )  /\  ( dom  R  u.  ran  R
)  =  ( dom 
R  u.  dom  R
) )  ->  U. U. R  =  ( dom  R  u.  dom  R ) )
6 unidm 3318 . . . . . . 7  |-  ( dom 
R  u.  dom  R
)  =  dom  R
75, 6syl6req 2332 . . . . . 6  |-  ( ( U. U. R  =  ( dom  R  u.  ran  R )  /\  ( dom  R  u.  ran  R
)  =  ( dom 
R  u.  dom  R
) )  ->  dom  R  =  U. U. R
)
87ex 423 . . . . 5  |-  ( U. U. R  =  ( dom 
R  u.  ran  R
)  ->  ( ( dom  R  u.  ran  R
)  =  ( dom 
R  u.  dom  R
)  ->  dom  R  = 
U. U. R ) )
94, 8syl5com 26 . . . 4  |-  ( dom 
R  =  ran  R  ->  ( U. U. R  =  ( dom  R  u.  ran  R )  ->  dom  R  =  U. U. R ) )
102, 9syl 15 . . 3  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  -> 
( U. U. R  =  ( dom  R  u.  ran  R )  ->  dom  R  =  U. U. R ) )
111, 10syl5com 26 . 2  |-  ( Rel 
R  ->  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  U. U. R ) )
1211anabsi5 790 1  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  U. U. R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623   A.wral 2543    u. cun 3150   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  preodom2  25226  dfps2  25289
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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