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Theorem fnelfp 25903
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 25902 . . 3  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
21eleq2d 2383 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =  x }
) )
3 fveq2 5563 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 19 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4eqeq12d 2330 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
65elrab3 2958 . 2  |-  ( X  e.  A  ->  ( X  e.  { x  e.  A  |  ( F `  x )  =  x }  <->  ( F `  X )  =  X ) )
72, 6sylan9bb 680 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 176    /\ wa 358    = wceq 1633    e. wcel 1701   {crab 2581    i^i cin 3185    _I cid 4341   dom cdm 4726    Fn wfn 5287   ` cfv 5292
This theorem is referenced by:  ismrcd1  25921  ismrcd2  25922  istopclsd  25923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-res 4738  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300
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