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Theorem elfix 24514
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 24471 . . 3  |-  Fix R  =  dom  ( R  i^i  _I  )
21eleq2i 2360 . 2  |-  ( A  e.  Fix R  <->  A  e.  dom  ( R  i^i  _I  ) )
3 elfix.1 . . . 4  |-  A  e. 
_V
43eldm 4892 . . 3  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x  A ( R  i^i  _I  ) x )
5 brin 4086 . . . . . 6  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  _I  x ) )
6 vex 2804 . . . . . . . 8  |-  x  e. 
_V
76ideq 4852 . . . . . . 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 1572 . . . 4  |-  ( E. x  A ( R  i^i  _I  ) x  <->  E. x ( A R x  /\  A  =  x ) )
11 breq2 4043 . . . . . . 7  |-  ( A  =  x  ->  ( A R A  <->  A R x ) )
1211biimparc 473 . . . . . 6  |-  ( ( A R x  /\  A  =  x )  ->  A R A )
1312exlimiv 1624 . . . . 5  |-  ( E. x ( A R x  /\  A  =  x )  ->  A R A )
14 eqid 2296 . . . . . 6  |-  A  =  A
15 breq2 4043 . . . . . . . 8  |-  ( x  =  A  ->  ( A R x  <->  A R A ) )
16 eqeq2 2305 . . . . . . . 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 2888 . . . . . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   class class class wbr 4039    _I cid 4320   dom cdm 4705   Fixcfix 24449
This theorem is referenced by:  elfix2  24515  dffix2  24516  fixcnv  24519  ellimits  24521  elfuns  24525  dfrdg4  24560  tfrqfree  24561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-dm 4715  df-fix 24471
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