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Theorem fnelnfp 26740
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 26739 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
21eleq2d 2505 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F 
\  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =/=  x }
) )
3 fveq2 5730 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 21 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4neeq12d 2618 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =/=  x  <->  ( F `  X )  =/=  X
) )
65elrab3 3095 . 2  |-  ( X  e.  A  ->  ( X  e.  { x  e.  A  |  ( F `  x )  =/=  x }  <->  ( F `  X )  =/=  X
) )
72, 6sylan9bb 682 1  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711    \ cdif 3319    _I cid 4495   dom cdm 4880    Fn wfn 5451   ` cfv 5456
This theorem is referenced by:  f1omvdmvd  27365  f1omvdconj  27368  f1otrspeq  27369  pmtrfinv  27381  symggen  27390  psgnunilem1  27395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464
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