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Theorem fndifnfp 6106
Description: Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
Assertion
Ref Expression
fndifnfp  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
Distinct variable groups:    x, F    x, A

Proof of Theorem fndifnfp
StepHypRef Expression
1 dffn2 5745 . . . . . . . 8  |-  ( F  Fn  A  <->  F : A
--> _V )
2 fssxp 5756 . . . . . . . 8  |-  ( F : A --> _V  ->  F 
C_  ( A  X.  _V ) )
31, 2sylbi 199 . . . . . . 7  |-  ( F  Fn  A  ->  F  C_  ( A  X.  _V ) )
4 ssdif0 3852 . . . . . . 7  |-  ( F 
C_  ( A  X.  _V )  <->  ( F  \ 
( A  X.  _V ) )  =  (/) )
53, 4sylib 200 . . . . . 6  |-  ( F  Fn  A  ->  ( F  \  ( A  X.  _V ) )  =  (/) )
65uneq2d 3621 . . . . 5  |-  ( F  Fn  A  ->  (
( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )  =  ( ( F  \  _I  )  u.  (/) ) )
7 un0 3788 . . . . 5  |-  ( ( F  \  _I  )  u.  (/) )  =  ( F  \  _I  )
86, 7syl6req 2481 . . . 4  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) ) )
9 df-res 4863 . . . . . 6  |-  (  _I  |`  A )  =  (  _I  i^i  ( A  X.  _V ) )
109difeq2i 3581 . . . . 5  |-  ( F 
\  (  _I  |`  A ) )  =  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )
11 difindi 3728 . . . . 5  |-  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
1210, 11eqtri 2452 . . . 4  |-  ( F 
\  (  _I  |`  A ) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
138, 12syl6eqr 2482 . . 3  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( F  \  (  _I  |`  A ) ) )
1413dmeqd 5054 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  dom  ( F  \ 
(  _I  |`  A ) ) )
15 fnresi 5709 . . 3  |-  (  _I  |`  A )  Fn  A
16 fndmdif 5999 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
1715, 16mpan2 676 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
18 fvresi 6103 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1918neeq2d 2703 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =/=  x
) )
2019rabbiia 3070 . . 3  |-  { x  e.  A  |  ( F `  x )  =/=  ( (  _I  |`  A ) `
 x ) }  =  { x  e.  A  |  ( F `
 x )  =/=  x }
2120a1i 11 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  =/=  ( (  _I  |`  A ) `
 x ) }  =  { x  e.  A  |  ( F `
 x )  =/=  x } )
2214, 17, 213eqtrd 2468 1  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869    =/= wne 2619   {crab 2780   _Vcvv 3082    \ cdif 3434    u. cun 3435    i^i cin 3436    C_ wss 3437   (/)c0 3762    _I cid 4761    X. cxp 4849   dom cdm 4851    |` cres 4853    Fn wfn 5594   -->wf 5595   ` cfv 5599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-fv 5607
This theorem is referenced by:  fnelnfp  6107  fnnfpeq0  6108  f1omvdcnv  17078  pmtrmvd  17090  pmtrdifellem4  17113  sygbasnfpfi  17146
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