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Theorem fninfp 26625
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 5123 . . . . . 6  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( (  _I  i^i  F )  |`  A )
2 incom 3493 . . . . . . 7  |-  (  _I 
i^i  F )  =  ( F  i^i  _I  )
32reseq1i 5101 . . . . . 6  |-  ( (  _I  i^i  F )  |`  A )  =  ( ( F  i^i  _I  )  |`  A )
41, 3eqtri 2424 . . . . 5  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
5 incom 3493 . . . . 5  |-  ( ( F  |`  A )  i^i  _I  )  =  (  _I  i^i  ( F  |`  A ) )
6 inres 5123 . . . . 5  |-  ( F  i^i  (  _I  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
74, 5, 63eqtr4i 2434 . . . 4  |-  ( ( F  |`  A )  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) )
8 fnresdm 5513 . . . . 5  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
98ineq1d 3501 . . . 4  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  _I  )  =  ( F  i^i  _I  ) )
107, 9syl5reqr 2451 . . 3  |-  ( F  Fn  A  ->  ( F  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) ) )
1110dmeqd 5031 . 2  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  dom  ( F  i^i  (  _I  |`  A ) ) )
12 fnresi 5521 . . 3  |-  (  _I  |`  A )  Fn  A
13 fndmin 5796 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  ( F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
1412, 13mpan2 653 . 2  |-  ( F  Fn  A  ->  dom  ( F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
15 fvresi 5883 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1615eqeq2d 2415 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
1716rabbiia 2906 . . 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 2440 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 1649    e. wcel 1721   {crab 2670    i^i cin 3279    _I cid 4453   dom cdm 4837    |` cres 4839    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  fnelfp  26626
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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