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Theorem cnvref2 25169
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 5214 . . . 4  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
21raleqdv 2755 . . 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 25168 . . 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 5067 . . . 4  |-  Rel  `' R
6 relfld 5214 . . . . 5  |-  ( Rel  `' R  ->  U. U. `' R  =  ( dom  `' R  u.  ran  `' R ) )
76eqcomd 2301 . . . 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 2755 . 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 1632   A.wral 2556    u. cun 3163   U.cuni 3843   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706   Rel wrel 4710
This theorem is referenced by:  relrefcnv  25220
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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