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Theorem cnvref2 25066
Description: The converse of a reflexive relation is reflexive. (Contributed by FL, 19-Sep-2011.)
Assertion
Ref Expression
cnvref2  |-  ( Rel 
R  ->  ( A. x  e.  U. U. R x R x  <->  A. x  e.  U. U. `' R x `' R x ) )
Distinct variable group:    x, R

Proof of Theorem cnvref2
StepHypRef Expression
1 relfld 5198 . . . 4  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
21raleqdv 2742 . . 3  |-  ( Rel 
R  ->  ( A. x  e.  U. U. R x R x  <->  A. x  e.  ( dom  R  u.  ran  R ) x R x ) )
3 cnvref 25065 . . 3  |-  ( A. x  e.  ( dom  R  u.  ran  R ) x R x  <->  A. x  e.  ( dom  `' R  u.  ran  `' R ) x `' R x )
42, 3syl6bb 252 . 2  |-  ( Rel 
R  ->  ( A. x  e.  U. U. R x R x  <->  A. x  e.  ( dom  `' R  u.  ran  `' R ) x `' R x ) )
5 relcnv 5051 . . . 4  |-  Rel  `' R
6 relfld 5198 . . . . 5  |-  ( Rel  `' R  ->  U. U. `' R  =  ( dom  `' R  u.  ran  `' R ) )
76eqcomd 2288 . . . 4  |-  ( Rel  `' R  ->  ( dom  `' R  u.  ran  `' R )  =  U. U. `' R )
85, 7mp1i 11 . . 3  |-  ( Rel 
R  ->  ( dom  `' R  u.  ran  `' R )  =  U. U. `' R )
98raleqdv 2742 . 2  |-  ( Rel 
R  ->  ( A. x  e.  ( dom  `' R  u.  ran  `' R ) x `' R x  <->  A. x  e.  U. U. `' R x `' R x ) )
104, 9bitrd 244 1  |-  ( Rel 
R  ->  ( A. x  e.  U. U. R x R x  <->  A. x  e.  U. U. `' R x `' R x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   A.wral 2543    u. cun 3150   U.cuni 3827   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ran crn 4690   Rel wrel 4694
This theorem is referenced by:  relrefcnv  25117
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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