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Theorem fnelnfp 6102
Description: Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
fnelnfp  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )

Proof of Theorem fnelnfp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fndifnfp 6101 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
21eleq2d 2537 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F 
\  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =/=  x }
) )
3 fveq2 5872 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 22 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4neeq12d 2746 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =/=  x  <->  ( F `  X )  =/=  X
) )
65elrab3 3267 . 2  |-  ( X  e.  A  ->  ( X  e.  { x  e.  A  |  ( F `  x )  =/=  x }  <->  ( F `  X )  =/=  X
) )
72, 6sylan9bb 699 1  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821    \ cdif 3478    _I cid 4796   dom cdm 5005    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  f1omvdmvd  16341  f1omvdconj  16344  f1otrspeq  16345  pmtrfinv  16359  symggen  16368  psgnunilem1  16391  mdetdiaglem  18969  mdetralt  18979  mdetunilem7  18989
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