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Theorem elfix 25740
Description: Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
elfix.1  |-  A  e. 
_V
Assertion
Ref Expression
elfix  |-  ( A  e.  Fix R  <->  A R A )

Proof of Theorem elfix
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fix 25695 . . 3  |-  Fix R  =  dom  ( R  i^i  _I  )
21eleq2i 2499 . 2  |-  ( A  e.  Fix R  <->  A  e.  dom  ( R  i^i  _I  ) )
3 elfix.1 . . . 4  |-  A  e. 
_V
43eldm 5059 . . 3  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x  A ( R  i^i  _I  ) x )
5 brin 4251 . . . . 5  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  _I  x ) )
6 ancom 438 . . . . 5  |-  ( ( A R x  /\  A  _I  x )  <->  ( A  _I  x  /\  A R x ) )
7 vex 2951 . . . . . . . 8  |-  x  e. 
_V
87ideq 5017 . . . . . . 7  |-  ( A  _I  x  <->  A  =  x )
9 eqcom 2437 . . . . . . 7  |-  ( A  =  x  <->  x  =  A )
108, 9bitri 241 . . . . . 6  |-  ( A  _I  x  <->  x  =  A )
1110anbi1i 677 . . . . 5  |-  ( ( A  _I  x  /\  A R x )  <->  ( x  =  A  /\  A R x ) )
125, 6, 113bitri 263 . . . 4  |-  ( A ( R  i^i  _I  ) x  <->  ( x  =  A  /\  A R x ) )
1312exbii 1592 . . 3  |-  ( E. x  A ( R  i^i  _I  ) x  <->  E. x ( x  =  A  /\  A R x ) )
144, 13bitri 241 . 2  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x
( x  =  A  /\  A R x ) )
15 breq2 4208 . . 3  |-  ( x  =  A  ->  ( A R x  <->  A R A ) )
163, 15ceqsexv 2983 . 2  |-  ( E. x ( x  =  A  /\  A R x )  <->  A R A )
172, 14, 163bitri 263 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   class class class wbr 4204    _I cid 4485   dom cdm 4870   Fixcfix 25671
This theorem is referenced by:  elfix2  25741  dffix2  25742  fixcnv  25745  ellimits  25747  elfuns  25752  dfrdg4  25787  tfrqfree  25788
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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