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Theorem relrefcnv 24529
Description: A relation is reflexive iff its converse is reflexive. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
relrefcnv  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  C_  R  <->  (  _I  |` 
U. U. `' R ) 
C_  `' R ) )

Proof of Theorem relrefcnv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 cnvref2 24478 . 2  |-  ( Rel 
R  ->  ( A. x  e.  U. U. R x R x  <->  A. x  e.  U. U. `' R x `' R x ) )
2 issref 5056 . 2  |-  ( (  _I  |`  U. U. R
)  C_  R  <->  A. x  e.  U. U. R x R x )
3 issref 5056 . 2  |-  ( (  _I  |`  U. U. `' R )  C_  `' R 
<-> 
A. x  e.  U. U. `' R x `' R x )
41, 2, 33bitr4g 279 1  |-  ( Rel 
R  ->  ( (  _I  |`  U. U. R
)  C_  R  <->  (  _I  |` 
U. U. `' R ) 
C_  `' R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wral 2543    C_ wss 3152   U.cuni 3827   class class class wbr 4023    _I cid 4304   `'ccnv 4688    |` cres 4691   Rel wrel 4694
This theorem is referenced by:  dupre1  24655
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-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701
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