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Theorem fndifnfp 26859
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 5406 . . . . . . . 8  |-  ( F  Fn  A  <->  F : A
--> _V )
2 fssxp 5416 . . . . . . . 8  |-  ( F : A --> _V  ->  F 
C_  ( A  X.  _V ) )
31, 2sylbi 187 . . . . . . 7  |-  ( F  Fn  A  ->  F  C_  ( A  X.  _V ) )
4 ssdif0 3526 . . . . . . 7  |-  ( F 
C_  ( A  X.  _V )  <->  ( F  \ 
( A  X.  _V ) )  =  (/) )
53, 4sylib 188 . . . . . 6  |-  ( F  Fn  A  ->  ( F  \  ( A  X.  _V ) )  =  (/) )
65uneq2d 3342 . . . . 5  |-  ( F  Fn  A  ->  (
( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )  =  ( ( F  \  _I  )  u.  (/) ) )
7 un0 3492 . . . . 5  |-  ( ( F  \  _I  )  u.  (/) )  =  ( F  \  _I  )
86, 7syl6req 2345 . . . 4  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) ) )
9 df-res 4717 . . . . . 6  |-  (  _I  |`  A )  =  (  _I  i^i  ( A  X.  _V ) )
109difeq2i 3304 . . . . 5  |-  ( F 
\  (  _I  |`  A ) )  =  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )
11 difindi 3436 . . . . 5  |-  ( F 
\  (  _I  i^i  ( A  X.  _V )
) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
1210, 11eqtri 2316 . . . 4  |-  ( F 
\  (  _I  |`  A ) )  =  ( ( F  \  _I  )  u.  ( F  \  ( A  X.  _V ) ) )
138, 12syl6eqr 2346 . . 3  |-  ( F  Fn  A  ->  ( F  \  _I  )  =  ( F  \  (  _I  |`  A ) ) )
1413dmeqd 4897 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  dom  ( F  \ 
(  _I  |`  A ) ) )
15 fnresi 5377 . . 3  |-  (  _I  |`  A )  Fn  A
16 fndmdif 5645 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
1715, 16mpan2 652 . 2  |-  ( F  Fn  A  ->  dom  ( F  \  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =/=  ( (  _I  |`  A ) `
 x ) } )
18 fvresi 5727 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1918neeq2d 2473 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =/=  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =/=  x
) )
2019rabbiia 2791 . . 3  |-  { x  e.  A  |  ( F `  x )  =/=  ( (  _I  |`  A ) `
 x ) }  =  { x  e.  A  |  ( F `
 x )  =/=  x }
2120a1i 10 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  =/=  ( (  _I  |`  A ) `
 x ) }  =  { x  e.  A  |  ( F `
 x )  =/=  x } )
2214, 17, 213eqtrd 2332 1  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468    _I cid 4320    X. cxp 4703   dom cdm 4705    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271
This theorem is referenced by:  fnelnfp  26860  fnnfpeq0  26861  f1omvdcnv  27490  pmtrmvd  27501
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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