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Theorem dffix2 25663
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 1684 . . . . 5  |-  x  =  x
2 vex 2923 . . . . . 6  |-  x  e. 
_V
3 breq1 4179 . . . . . . 7  |-  ( y  =  x  ->  (
y A x  <->  x A x ) )
4 equequ1 1692 . . . . . . 7  |-  ( y  =  x  ->  (
y  =  x  <->  x  =  x ) )
53, 4anbi12d 692 . . . . . 6  |-  ( y  =  x  ->  (
( y A x  /\  y  =  x )  <->  ( x A x  /\  x  =  x ) ) )
62, 5spcev 3007 . . . . 5  |-  ( ( x A x  /\  x  =  x )  ->  E. y ( y A x  /\  y  =  x ) )
71, 6mpan2 653 . . . 4  |-  ( x A x  ->  E. y
( y A x  /\  y  =  x ) )
83biimpac 473 . . . . 5  |-  ( ( y A x  /\  y  =  x )  ->  x A x )
98exlimiv 1641 . . . 4  |-  ( E. y ( y A x  /\  y  =  x )  ->  x A x )
107, 9impbii 181 . . 3  |-  ( x A x  <->  E. y
( y A x  /\  y  =  x ) )
112elfix 25661 . . 3  |-  ( x  e.  Fix A  <->  x A x )
122elrn 5073 . . . 4  |-  ( x  e.  ran  ( A  i^i  _I  )  <->  E. y 
y ( A  i^i  _I  ) x )
13 brin 4223 . . . . . 6  |-  ( y ( A  i^i  _I  ) x  <->  ( y A x  /\  y  _I  x ) )
142ideq 4988 . . . . . . 7  |-  ( y  _I  x  <->  y  =  x )
1514anbi2i 676 . . . . . 6  |-  ( ( y A x  /\  y  _I  x )  <->  ( y A x  /\  y  =  x )
)
1613, 15bitri 241 . . . . 5  |-  ( y ( A  i^i  _I  ) x  <->  ( y A x  /\  y  =  x ) )
1716exbii 1589 . . . 4  |-  ( E. y  y ( A  i^i  _I  ) x  <->  E. y ( y A x  /\  y  =  x ) )
1812, 17bitri 241 . . 3  |-  ( x  e.  ran  ( A  i^i  _I  )  <->  E. y
( y A x  /\  y  =  x ) )
1910, 11, 183bitr4i 269 . 2  |-  ( x  e.  Fix A  <->  x  e.  ran  ( A  i^i  _I  ) )
2019eqriv 2405 1  |-  Fix A  =  ran  ( A  i^i  _I  )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    i^i cin 3283   class class class wbr 4176    _I cid 4457   ran crn 4842   Fixcfix 25596
This theorem is referenced by:  fixssrn  25665
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-dm 4851  df-rn 4852  df-fix 25618
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