Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fndifnfp Unicode version

Theorem fndifnfp 26428
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 5532 . . . . . . . 8  |-  ( F  Fn  A  <->  F : A
--> _V )
2 fssxp 5542 . . . . . . . 8  |-  ( F : A --> _V  ->  F 
C_  ( A  X.  _V ) )
31, 2sylbi 188 . . . . . . 7  |-  ( F  Fn  A  ->  F  C_  ( A  X.  _V ) )
4 ssdif0 3629 . . . . . . 7  |-  ( F 
C_  ( A  X.  _V )  <->  ( F  \ 
( A  X.  _V ) )  =  (/) )
53, 4sylib 189 . . . . . 6  |-  ( F  Fn  A  ->  ( F  \  ( A  X.  _V ) )  =  (/) )
65uneq2d 3444 . . . . 5  |-  ( F  Fn  A  ->  (
( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )  =  ( ( F  \  _I  )  u.  (/) ) )
7 un0 3595 . . . . 5  |-  ( ( F  \  _I  )  u.  (/) )  =  ( F  \  _I  )
86, 7syl6req 2436 . . . 4  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) ) )
9 df-res 4830 . . . . . 6  |-  (  _I  |`  A )  =  (  _I  i^i  ( A  X.  _V ) )
109difeq2i 3405 . . . . 5  |-  ( F 
\  (  _I  |`  A ) )  =  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )
11 difindi 3538 . . . . 5  |-  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
1210, 11eqtri 2407 . . . 4  |-  ( F 
\  (  _I  |`  A ) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
138, 12syl6eqr 2437 . . 3  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( F  \  (  _I  |`  A ) ) )
1413dmeqd 5012 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  dom  ( F  \ 
(  _I  |`  A ) ) )
15 fnresi 5502 . . 3  |-  (  _I  |`  A )  Fn  A
16 fndmdif 5773 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
1715, 16mpan2 653 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
18 fvresi 5863 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1918neeq2d 2564 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =/=  x
) )
2019rabbiia 2889 . . 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 2423 1  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2550   {crab 2653   _Vcvv 2899    \ cdif 3260    u. cun 3261    i^i cin 3262    C_ wss 3263   (/)c0 3571    _I cid 4434    X. cxp 4816   dom cdm 4818    |` cres 4820    Fn wfn 5389   -->wf 5390   ` cfv 5394
This theorem is referenced by:  fnelnfp  26429  fnnfpeq0  26430  f1omvdcnv  27056  pmtrmvd  27067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402
  Copyright terms: Public domain W3C validator