Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dffix2 Unicode version

Theorem dffix2 25186
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 1681 . . . . 5  |-  x  =  x
2 vex 2876 . . . . . 6  |-  x  e. 
_V
3 breq1 4128 . . . . . . 7  |-  ( y  =  x  ->  (
y A x  <->  x A x ) )
4 equequ1 1689 . . . . . . 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 2960 . . . . 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 1639 . . . 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 25184 . . 3  |-  ( x  e.  Fix A  <->  x A x )
122elrn 5022 . . . 4  |-  ( x  e.  ran  ( A  i^i  _I  )  <->  E. y 
y ( A  i^i  _I  ) x )
13 brin 4172 . . . . . 6  |-  ( y ( A  i^i  _I  ) x  <->  ( y A x  /\  y  _I  x ) )
142ideq 4939 . . . . . . 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 1587 . . . 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 2363 1  |-  Fix A  =  ran  ( A  i^i  _I  )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1546    = wceq 1647    e. wcel 1715    i^i cin 3237   class class class wbr 4125    _I cid 4407   ran crn 4793   Fixcfix 25119
This theorem is referenced by:  fixssrn  25188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-fix 25141
  Copyright terms: Public domain W3C validator