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Theorem cnvref 25065
Description: The converse of a reflexive class is reflexive. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
cnvref  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  <->  A. x  e.  ( dom  `' R  u.  ran  `' R ) x `' R x )
Distinct variable group:    x, R

Proof of Theorem cnvref
StepHypRef Expression
1 brcnvg 4862 . . . . 5  |-  ( ( x  e.  ( dom 
R  u.  ran  R
)  /\  x  e.  ( dom  R  u.  ran  R ) )  ->  (
x `' R x  <-> 
x R x ) )
21bicomd 192 . . . 4  |-  ( ( x  e.  ( dom 
R  u.  ran  R
)  /\  x  e.  ( dom  R  u.  ran  R ) )  ->  (
x R x  <->  x `' R x ) )
32anidms 626 . . 3  |-  ( x  e.  ( dom  R  u.  ran  R )  -> 
( x R x  <-> 
x `' R x ) )
43ralbiia 2575 . 2  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  <->  A. x  e.  ( dom  R  u.  ran  R ) x `' R x )
5 fldcnv 25056 . . 3  |-  ( dom 
R  u.  ran  R
)  =  ( dom  `' R  u.  ran  `' R )
65raleqi 2740 . 2  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x `' R x  <->  A. x  e.  ( dom  `' R  u.  ran  `' R ) x `' R x )
74, 6bitri 240 1  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  <->  A. x  e.  ( dom  `' R  u.  ran  `' R ) x `' R x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   A.wral 2543    u. cun 3150   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690
This theorem is referenced by:  cnvref2  25066
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-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-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700
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