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Theorem fninfp 26857
Description: Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fninfp  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
Distinct variable groups:    x, F    x, A

Proof of Theorem fninfp
StepHypRef Expression
1 inres 4989 . . . . . 6  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( (  _I  i^i  F )  |`  A )
2 incom 3374 . . . . . . 7  |-  (  _I 
i^i  F )  =  ( F  i^i  _I  )
32reseq1i 4967 . . . . . 6  |-  ( (  _I  i^i  F )  |`  A )  =  ( ( F  i^i  _I  )  |`  A )
41, 3eqtri 2316 . . . . 5  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
5 incom 3374 . . . . 5  |-  ( ( F  |`  A )  i^i  _I  )  =  (  _I  i^i  ( F  |`  A ) )
6 inres 4989 . . . . 5  |-  ( F  i^i  (  _I  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
74, 5, 63eqtr4i 2326 . . . 4  |-  ( ( F  |`  A )  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) )
8 fnresdm 5369 . . . . 5  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
98ineq1d 3382 . . . 4  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  _I  )  =  ( F  i^i  _I  ) )
107, 9syl5reqr 2343 . . 3  |-  ( F  Fn  A  ->  ( F  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) ) )
1110dmeqd 4897 . 2  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  dom  ( F  i^i  (  _I  |`  A ) ) )
12 fnresi 5377 . . 3  |-  (  _I  |`  A )  Fn  A
13 fndmin 5648 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
1412, 13mpan2 652 . 2  |-  ( F  Fn  A  ->  dom  ( F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
15 fvresi 5727 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1615eqeq2d 2307 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
1716rabbiia 2791 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =  x }
1817a1i 10 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =  x }
)
1911, 14, 183eqtrd 2332 1  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560    i^i cin 3164    _I cid 4320   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  fnelfp  26858
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-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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