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Theorem domfldref 25164
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 5214 . . 3  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
2 domrngref 25163 . . . 4  |-  ( ( Rel  R  /\  A. x  e.  U. U. R x R x )  ->  dom  R  =  ran  R
)
3 uneq2 3336 . . . . . 6  |-  ( dom 
R  =  ran  R  ->  ( dom  R  u.  dom  R )  =  ( dom  R  u.  ran  R ) )
43eqcomd 2301 . . . . 5  |-  ( dom 
R  =  ran  R  ->  ( dom  R  u.  ran  R )  =  ( dom  R  u.  dom  R ) )
5 eqtr 2313 . . . . . . 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 3331 . . . . . . 7  |-  ( dom 
R  u.  dom  R
)  =  dom  R
75, 6syl6req 2345 . . . . . 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 1632   A.wral 2556    u. cun 3163   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ran crn 4706   Rel wrel 4710
This theorem is referenced by:  preodom2  25329  dfps2  25392
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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