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Theorem elfix 24443
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 24400 . . 3  |-  Fix R  =  dom  ( R  i^i  _I  )
21eleq2i 2347 . 2  |-  ( A  e.  Fix R  <->  A  e.  dom  ( R  i^i  _I  ) )
3 elfix.1 . . . 4  |-  A  e. 
_V
43eldm 4876 . . 3  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x  A ( R  i^i  _I  ) x )
5 brin 4070 . . . . . 6  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  _I  x ) )
6 vex 2791 . . . . . . . 8  |-  x  e. 
_V
76ideq 4836 . . . . . . 7  |-  ( A  _I  x  <->  A  =  x )
87anbi2i 675 . . . . . 6  |-  ( ( A R x  /\  A  _I  x )  <->  ( A R x  /\  A  =  x )
)
95, 8bitri 240 . . . . 5  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  =  x ) )
109exbii 1569 . . . 4  |-  ( E. x  A ( R  i^i  _I  ) x  <->  E. x ( A R x  /\  A  =  x ) )
11 breq2 4027 . . . . . . 7  |-  ( A  =  x  ->  ( A R A  <->  A R x ) )
1211biimparc 473 . . . . . 6  |-  ( ( A R x  /\  A  =  x )  ->  A R A )
1312exlimiv 1666 . . . . 5  |-  ( E. x ( A R x  /\  A  =  x )  ->  A R A )
14 eqid 2283 . . . . . 6  |-  A  =  A
15 breq2 4027 . . . . . . . 8  |-  ( x  =  A  ->  ( A R x  <->  A R A ) )
16 eqeq2 2292 . . . . . . . 8  |-  ( x  =  A  ->  ( A  =  x  <->  A  =  A ) )
1715, 16anbi12d 691 . . . . . . 7  |-  ( x  =  A  ->  (
( A R x  /\  A  =  x )  <->  ( A R A  /\  A  =  A ) ) )
183, 17spcev 2875 . . . . . 6  |-  ( ( A R A  /\  A  =  A )  ->  E. x ( A R x  /\  A  =  x ) )
1914, 18mpan2 652 . . . . 5  |-  ( A R A  ->  E. x
( A R x  /\  A  =  x ) )
2013, 19impbii 180 . . . 4  |-  ( E. x ( A R x  /\  A  =  x )  <->  A R A )
2110, 20bitri 240 . . 3  |-  ( E. x  A ( R  i^i  _I  ) x  <-> 
A R A )
224, 21bitri 240 . 2  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  A R A )
232, 22bitri 240 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   class class class wbr 4023    _I cid 4304   dom cdm 4689   Fixcfix 24378
This theorem is referenced by:  elfix2  24444  dffix2  24445  fixcnv  24448  ellimits  24450  elfuns  24454  dfrdg4  24488  tfrqfree  24489
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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