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Theorem fnelfp 26736
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 26735 . . 3  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
21eleq2d 2503 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =  x }
) )
3 fveq2 5728 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 20 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4eqeq12d 2450 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
65elrab3 3093 . 2  |-  ( X  e.  A  ->  ( X  e.  { x  e.  A  |  ( F `  x )  =  x }  <->  ( F `  X )  =  X ) )
72, 6sylan9bb 681 1  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  ( F `  X )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    i^i cin 3319    _I cid 4493   dom cdm 4878    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  ismrcd1  26752  ismrcd2  26753  istopclsd  26754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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