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Theorem elfix2 25470
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 2909 . 2  |-  ( A  e.  Fix R  ->  A  e.  _V )
2 elfix2.1 . . 3  |-  Rel  R
32brrelexi 4860 . 2  |-  ( A R A  ->  A  e.  _V )
4 eleq1 2449 . . 3  |-  ( x  =  A  ->  (
x  e.  Fix R  <->  A  e.  Fix R ) )
5 breq12 4160 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
65anidms 627 . . 3  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
7 vex 2904 . . . 4  |-  x  e. 
_V
87elfix 25469 . . 3  |-  ( x  e.  Fix R  <->  x R x )
94, 6, 8vtoclbg 2957 . 2  |-  ( A  e.  _V  ->  ( A  e.  Fix R  <->  A R A ) )
101, 3, 9pm5.21nii 343 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2901   class class class wbr 4155   Rel wrel 4825   Fixcfix 25404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-dm 4830  df-fix 25426
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