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Theorem elfix2 25741
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 2956 . 2  |-  ( A  e.  Fix R  ->  A  e.  _V )
2 elfix2.1 . . 3  |-  Rel  R
32brrelexi 4910 . 2  |-  ( A R A  ->  A  e.  _V )
4 eleq1 2495 . . 3  |-  ( x  =  A  ->  (
x  e.  Fix R  <->  A  e.  Fix R ) )
5 breq12 4209 . . . 4  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
65anidms 627 . . 3  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
7 vex 2951 . . . 4  |-  x  e. 
_V
87elfix 25740 . . 3  |-  ( x  e.  Fix R  <->  x R x )
94, 6, 8vtoclbg 3004 . 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 1652    e. wcel 1725   _Vcvv 2948   class class class wbr 4204   Rel wrel 4875   Fixcfix 25671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-dm 4880  df-fix 25695
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