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Theorem domfldrefc 25057
Description: The domain of a reflexive class equals its field. (Contributed by FL, 29-Dec-2011.)
Assertion
Ref Expression
domfldrefc  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ( dom 
R  u.  ran  R
) )
Distinct variable group:    x, R

Proof of Theorem domfldrefc
StepHypRef Expression
1 vex 2791 . . . . . . 7  |-  x  e. 
_V
21, 1breldm 4883 . . . . . 6  |-  ( x R x  ->  x  e.  dom  R )
32ralimi 2618 . . . . 5  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  A. x  e.  ( dom  R  u.  ran  R
) x  e.  dom  R )
4 df-ral 2548 . . . . 5  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x  e.  dom  R  <->  A. x ( x  e.  ( dom  R  u.  ran  R )  ->  x  e.  dom  R ) )
53, 4sylib 188 . . . 4  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  A. x ( x  e.  ( dom  R  u.  ran  R )  ->  x  e.  dom  R ) )
6 dfss2 3169 . . . 4  |-  ( ( dom  R  u.  ran  R )  C_  dom  R  <->  A. x
( x  e.  ( dom  R  u.  ran  R )  ->  x  e.  dom  R ) )
75, 6sylibr 203 . . 3  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  -> 
( dom  R  u.  ran  R )  C_  dom  R )
8 ssun1 3338 . . 3  |-  dom  R  C_  ( dom  R  u.  ran  R )
97, 8jctil 523 . 2  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  -> 
( dom  R  C_  ( dom  R  u.  ran  R
)  /\  ( dom  R  u.  ran  R ) 
C_  dom  R )
)
10 eqss 3194 . 2  |-  ( dom 
R  =  ( dom 
R  u.  ran  R
)  <->  ( dom  R  C_  ( dom  R  u.  ran  R )  /\  ( dom  R  u.  ran  R
)  C_  dom  R ) )
119, 10sylibr 203 1  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  ->  dom  R  =  ( dom 
R  u.  ran  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543    u. cun 3150    C_ wss 3152   class class class wbr 4023   dom cdm 4689   ran crn 4690
This theorem is referenced by:  dranfldrefc  25059
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  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-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-dm 4699
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