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Theorem fndifnfp 6080
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 5715 . . . . . . . 8  |-  ( F  Fn  A  <->  F : A
--> _V )
2 fssxp 5726 . . . . . . . 8  |-  ( F : A --> _V  ->  F 
C_  ( A  X.  _V ) )
31, 2sylbi 195 . . . . . . 7  |-  ( F  Fn  A  ->  F  C_  ( A  X.  _V ) )
4 ssdif0 3828 . . . . . . 7  |-  ( F 
C_  ( A  X.  _V )  <->  ( F  \ 
( A  X.  _V ) )  =  (/) )
53, 4sylib 196 . . . . . 6  |-  ( F  Fn  A  ->  ( F  \  ( A  X.  _V ) )  =  (/) )
65uneq2d 3597 . . . . 5  |-  ( F  Fn  A  ->  (
( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )  =  ( ( F  \  _I  )  u.  (/) ) )
7 un0 3764 . . . . 5  |-  ( ( F  \  _I  )  u.  (/) )  =  ( F  \  _I  )
86, 7syl6req 2460 . . . 4  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) ) )
9 df-res 4835 . . . . . 6  |-  (  _I  |`  A )  =  (  _I  i^i  ( A  X.  _V ) )
109difeq2i 3558 . . . . 5  |-  ( F 
\  (  _I  |`  A ) )  =  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )
11 difindi 3704 . . . . 5  |-  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
1210, 11eqtri 2431 . . . 4  |-  ( F 
\  (  _I  |`  A ) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
138, 12syl6eqr 2461 . . 3  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( F  \  (  _I  |`  A ) ) )
1413dmeqd 5026 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  dom  ( F  \ 
(  _I  |`  A ) ) )
15 fnresi 5679 . . 3  |-  (  _I  |`  A )  Fn  A
16 fndmdif 5969 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
1715, 16mpan2 669 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
18 fvresi 6077 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1918neeq2d 2681 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =/=  x
) )
2019rabbiia 3048 . . 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 2447 1  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2758   _Vcvv 3059    \ cdif 3411    u. cun 3412    i^i cin 3413    C_ wss 3414   (/)c0 3738    _I cid 4733    X. cxp 4821   dom cdm 4823    |` cres 4825    Fn wfn 5564   -->wf 5565   ` cfv 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fv 5577
This theorem is referenced by:  fnelnfp  6081  fnnfpeq0  6082  f1omvdcnv  16793  pmtrmvd  16805  pmtrdifellem4  16828  sygbasnfpfi  16861
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