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Theorem fnelfp 6001
Description: Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnelfp  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  ( F `  X )  =  X ) )

Proof of Theorem fnelfp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fninfp 6000 . . 3  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
21eleq2d 2452 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =  x }
) )
3 fveq2 5774 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 22 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4eqeq12d 2404 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
65elrab3 3183 . 2  |-  ( X  e.  A  ->  ( X  e.  { x  e.  A  |  ( F `  x )  =  x }  <->  ( F `  X )  =  X ) )
72, 6sylan9bb 697 1  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  ( F `  X )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   {crab 2736    i^i cin 3388    _I cid 4704   dom cdm 4913    Fn wfn 5491   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504
This theorem is referenced by:  ismrcd1  30796  ismrcd2  30797  istopclsd  30798
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