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Theorem fnelnfp 26860
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 26859 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
21eleq2d 2363 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F 
\  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =/=  x }
) )
3 fveq2 5541 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 19 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4neeq12d 2474 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =/=  x  <->  ( F `  X )  =/=  X
) )
65elrab3 2937 . 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  )  <->  ( F `  X )  =/=  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162    _I cid 4320   dom cdm 4705    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  f1omvdmvd  27489  f1omvdconj  27492  f1otrspeq  27493  pmtrfinv  27505  symggen  27514  psgnunilem1  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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