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Theorem dffix2 24445
Description: The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
dffix2  |-  Fix A  =  ran  ( A  i^i  _I  )

Proof of Theorem dffix2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equid 1644 . . . . 5  |-  x  =  x
2 vex 2791 . . . . . 6  |-  x  e. 
_V
3 breq1 4026 . . . . . . 7  |-  ( y  =  x  ->  (
y A x  <->  x A x ) )
4 equequ1 1648 . . . . . . 7  |-  ( y  =  x  ->  (
y  =  x  <->  x  =  x ) )
53, 4anbi12d 691 . . . . . 6  |-  ( y  =  x  ->  (
( y A x  /\  y  =  x )  <->  ( x A x  /\  x  =  x ) ) )
62, 5spcev 2875 . . . . 5  |-  ( ( x A x  /\  x  =  x )  ->  E. y ( y A x  /\  y  =  x ) )
71, 6mpan2 652 . . . 4  |-  ( x A x  ->  E. y
( y A x  /\  y  =  x ) )
83biimpac 472 . . . . 5  |-  ( ( y A x  /\  y  =  x )  ->  x A x )
98exlimiv 1666 . . . 4  |-  ( E. y ( y A x  /\  y  =  x )  ->  x A x )
107, 9impbii 180 . . 3  |-  ( x A x  <->  E. y
( y A x  /\  y  =  x ) )
112elfix 24443 . . 3  |-  ( x  e.  Fix A  <->  x A x )
122elrn 4919 . . . 4  |-  ( x  e.  ran  ( A  i^i  _I  )  <->  E. y 
y ( A  i^i  _I  ) x )
13 brin 4070 . . . . . 6  |-  ( y ( A  i^i  _I  ) x  <->  ( y A x  /\  y  _I  x ) )
142ideq 4836 . . . . . . 7  |-  ( y  _I  x  <->  y  =  x )
1514anbi2i 675 . . . . . 6  |-  ( ( y A x  /\  y  _I  x )  <->  ( y A x  /\  y  =  x )
)
1613, 15bitri 240 . . . . 5  |-  ( y ( A  i^i  _I  ) x  <->  ( y A x  /\  y  =  x ) )
1716exbii 1569 . . . 4  |-  ( E. y  y ( A  i^i  _I  ) x  <->  E. y ( y A x  /\  y  =  x ) )
1812, 17bitri 240 . . 3  |-  ( x  e.  ran  ( A  i^i  _I  )  <->  E. y
( y A x  /\  y  =  x ) )
1910, 11, 183bitr4i 268 . 2  |-  ( x  e.  Fix A  <->  x  e.  ran  ( A  i^i  _I  ) )
2019eqriv 2280 1  |-  Fix A  =  ran  ( A  i^i  _I  )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    i^i cin 3151   class class class wbr 4023    _I cid 4304   ran crn 4690   Fixcfix 24378
This theorem is referenced by:  fixssrn  24447
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-cnv 4697  df-dm 4699  df-rn 4700  df-fix 24400
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