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Theorem fndifnfp 5905
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 5558 . . . . . . . 8  |-  ( F  Fn  A  <->  F : A
--> _V )
2 fssxp 5568 . . . . . . . 8  |-  ( F : A --> _V  ->  F 
C_  ( A  X.  _V ) )
31, 2sylbi 195 . . . . . . 7  |-  ( F  Fn  A  ->  F  C_  ( A  X.  _V ) )
4 ssdif0 3735 . . . . . . 7  |-  ( F 
C_  ( A  X.  _V )  <->  ( F  \ 
( A  X.  _V ) )  =  (/) )
53, 4sylib 196 . . . . . 6  |-  ( F  Fn  A  ->  ( F  \  ( A  X.  _V ) )  =  (/) )
65uneq2d 3508 . . . . 5  |-  ( F  Fn  A  ->  (
( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )  =  ( ( F  \  _I  )  u.  (/) ) )
7 un0 3660 . . . . 5  |-  ( ( F  \  _I  )  u.  (/) )  =  ( F  \  _I  )
86, 7syl6req 2490 . . . 4  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) ) )
9 df-res 4850 . . . . . 6  |-  (  _I  |`  A )  =  (  _I  i^i  ( A  X.  _V ) )
109difeq2i 3469 . . . . 5  |-  ( F 
\  (  _I  |`  A ) )  =  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )
11 difindi 3602 . . . . 5  |-  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
1210, 11eqtri 2461 . . . 4  |-  ( F 
\  (  _I  |`  A ) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
138, 12syl6eqr 2491 . . 3  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( F  \  (  _I  |`  A ) ) )
1413dmeqd 5040 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  dom  ( F  \ 
(  _I  |`  A ) ) )
15 fnresi 5526 . . 3  |-  (  _I  |`  A )  Fn  A
16 fndmdif 5805 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
1715, 16mpan2 671 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
18 fvresi 5902 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1918neeq2d 2620 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =/=  x
) )
2019rabbiia 2959 . . 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 2477 1  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2604   {crab 2717   _Vcvv 2970    \ cdif 3323    u. cun 3324    i^i cin 3325    C_ wss 3326   (/)c0 3635    _I cid 4629    X. cxp 4836   dom cdm 4838    |` cres 4840    Fn wfn 5411   -->wf 5412   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424
This theorem is referenced by:  fnelnfp  5906  fnnfpeq0  5907  f1omvdcnv  15948  pmtrmvd  15960  pmtrdifellem4  15983  sygbasnfpfi  16016  zrhcofipsgn  18021
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