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Theorem elfix 30682
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 30637 . . 3  |-  Fix R  =  dom  ( R  i^i  _I  )
21eleq2i 2523 . 2  |-  ( A  e.  Fix R  <->  A  e.  dom  ( R  i^i  _I  ) )
3 elfix.1 . . . 4  |-  A  e. 
_V
43eldm 5035 . . 3  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x  A ( R  i^i  _I  ) x )
5 brin 4455 . . . . 5  |-  ( A ( R  i^i  _I  ) x  <->  ( A R x  /\  A  _I  x ) )
6 ancom 452 . . . . 5  |-  ( ( A R x  /\  A  _I  x )  <->  ( A  _I  x  /\  A R x ) )
7 vex 3050 . . . . . . . 8  |-  x  e. 
_V
87ideq 4990 . . . . . . 7  |-  ( A  _I  x  <->  A  =  x )
9 eqcom 2460 . . . . . . 7  |-  ( A  =  x  <->  x  =  A )
108, 9bitri 253 . . . . . 6  |-  ( A  _I  x  <->  x  =  A )
1110anbi1i 702 . . . . 5  |-  ( ( A  _I  x  /\  A R x )  <->  ( x  =  A  /\  A R x ) )
125, 6, 113bitri 275 . . . 4  |-  ( A ( R  i^i  _I  ) x  <->  ( x  =  A  /\  A R x ) )
1312exbii 1720 . . 3  |-  ( E. x  A ( R  i^i  _I  ) x  <->  E. x ( x  =  A  /\  A R x ) )
144, 13bitri 253 . 2  |-  ( A  e.  dom  ( R  i^i  _I  )  <->  E. x
( x  =  A  /\  A R x ) )
15 breq2 4409 . . 3  |-  ( x  =  A  ->  ( A R x  <->  A R A ) )
163, 15ceqsexv 3086 . 2  |-  ( E. x ( x  =  A  /\  A R x )  <->  A R A )
172, 14, 163bitri 275 1  |-  ( A  e.  Fix R  <->  A R A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889   _Vcvv 3047    i^i cin 3405   class class class wbr 4405    _I cid 4747   dom cdm 4837   Fixcfix 30613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-br 4406  df-opab 4465  df-id 4752  df-xp 4843  df-rel 4844  df-dm 4847  df-fix 30637
This theorem is referenced by:  elfix2  30683  dffix2  30684  fixcnv  30687  ellimits  30689  elfuns  30694  dfrecs2  30729  dfrdg4  30730
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