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Theorem fninfp 5905
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 5128 . . . . . 6  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( (  _I  i^i  F )  |`  A )
2 incom 3543 . . . . . . 7  |-  (  _I 
i^i  F )  =  ( F  i^i  _I  )
32reseq1i 5106 . . . . . 6  |-  ( (  _I  i^i  F )  |`  A )  =  ( ( F  i^i  _I  )  |`  A )
41, 3eqtri 2463 . . . . 5  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
5 incom 3543 . . . . 5  |-  ( ( F  |`  A )  i^i  _I  )  =  (  _I  i^i  ( F  |`  A ) )
6 inres 5128 . . . . 5  |-  ( F  i^i  (  _I  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
74, 5, 63eqtr4i 2473 . . . 4  |-  ( ( F  |`  A )  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) )
8 fnresdm 5520 . . . . 5  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
98ineq1d 3551 . . . 4  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  _I  )  =  ( F  i^i  _I  ) )
107, 9syl5reqr 2490 . . 3  |-  ( F  Fn  A  ->  ( F  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) ) )
1110dmeqd 5042 . 2  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  dom  ( F  i^i  (  _I  |`  A ) ) )
12 fnresi 5528 . . 3  |-  (  _I  |`  A )  Fn  A
13 fndmin 5810 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
1412, 13mpan2 671 . 2  |-  ( F  Fn  A  ->  dom  ( F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
15 fvresi 5904 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1615eqeq2d 2454 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
1716rabbiia 2961 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =  x }
1817a1i 11 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =  x }
)
1911, 14, 183eqtrd 2479 1  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2719    i^i cin 3327    _I cid 4631   dom cdm 4840    |` cres 4842    Fn wfn 5413   ` cfv 5418
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 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-res 4852  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426
This theorem is referenced by:  fnelfp  5906
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