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Theorem elfix 25460
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 25417 . . 3  |-  Fix R  =  dom  ( R  i^i  _I  )
21eleq2i 2444 . 2  |-  ( A  e.  Fix R  <->  A  e.  dom  ( R  i^i  _I  ) )
3 elfix.1 . . . 4  |-  A  e. 
_V
43eldm 5000 . . 3  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x  A ( R  i^i  _I  ) x )
5 brin 4193 . . . . . 6  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  _I  x ) )
6 vex 2895 . . . . . . . 8  |-  x  e. 
_V
76ideq 4958 . . . . . . 7  |-  ( A  _I  x  <->  A  =  x )
87anbi2i 676 . . . . . 6  |-  ( ( A R x  /\  A  _I  x )  <->  ( A R x  /\  A  =  x )
)
95, 8bitri 241 . . . . 5  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  =  x ) )
109exbii 1589 . . . 4  |-  ( E. x  A ( R  i^i  _I  ) x  <->  E. x ( A R x  /\  A  =  x ) )
11 breq2 4150 . . . . . . 7  |-  ( A  =  x  ->  ( A R A  <->  A R x ) )
1211biimparc 474 . . . . . 6  |-  ( ( A R x  /\  A  =  x )  ->  A R A )
1312exlimiv 1641 . . . . 5  |-  ( E. x ( A R x  /\  A  =  x )  ->  A R A )
14 eqid 2380 . . . . . 6  |-  A  =  A
15 breq2 4150 . . . . . . . 8  |-  ( x  =  A  ->  ( A R x  <->  A R A ) )
16 eqeq2 2389 . . . . . . . 8  |-  ( x  =  A  ->  ( A  =  x  <->  A  =  A ) )
1715, 16anbi12d 692 . . . . . . 7  |-  ( x  =  A  ->  (
( A R x  /\  A  =  x )  <->  ( A R A  /\  A  =  A ) ) )
183, 17spcev 2979 . . . . . 6  |-  ( ( A R A  /\  A  =  A )  ->  E. x ( A R x  /\  A  =  x ) )
1914, 18mpan2 653 . . . . 5  |-  ( A R A  ->  E. x
( A R x  /\  A  =  x ) )
2013, 19impbii 181 . . . 4  |-  ( E. x ( A R x  /\  A  =  x )  <->  A R A )
2110, 20bitri 241 . . 3  |-  ( E. x  A ( R  i^i  _I  ) x  <-> 
A R A )
224, 21bitri 241 . 2  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  A R A )
232, 22bitri 241 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2892    i^i cin 3255   class class class wbr 4146    _I cid 4427   dom cdm 4811   Fixcfix 25395
This theorem is referenced by:  elfix2  25461  dffix2  25462  fixcnv  25465  ellimits  25467  elfuns  25471  dfrdg4  25506  tfrqfree  25507
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 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-dm 4821  df-fix 25417
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