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Theorem elfix2 24444
Description: Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
elfix2.1  |-  Rel  R
Assertion
Ref Expression
elfix2  |-  ( A  e.  Fix R  <->  A R A )

Proof of Theorem elfix2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  Fix R  ->  A  e.  _V )
2 elfix2.1 . . 3  |-  Rel  R
32brrelexi 4729 . 2  |-  ( A R A  ->  A  e.  _V )
4 eleq1 2343 . . 3  |-  ( x  =  A  ->  (
x  e.  Fix R  <->  A  e.  Fix R ) )
5 breq12 4028 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
65anidms 626 . . 3  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
7 vex 2791 . . . 4  |-  x  e. 
_V
87elfix 24443 . . 3  |-  ( x  e.  Fix R  <->  x R x )
94, 6, 8vtoclbg 2844 . 2  |-  ( A  e.  _V  ->  ( A  e.  Fix R  <->  A R A ) )
101, 3, 9pm5.21nii 342 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   Rel wrel 4694   Fixcfix 24378
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-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-dm 4699  df-fix 24400
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