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Theorem fninfp 25920
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 4880 . . . . . 6  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( (  _I  i^i  F )  |`  A )
2 incom 3269 . . . . . . 7  |-  (  _I 
i^i  F )  =  ( F  i^i  _I  )
32reseq1i 4858 . . . . . 6  |-  ( (  _I  i^i  F )  |`  A )  =  ( ( F  i^i  _I  )  |`  A )
41, 3eqtri 2273 . . . . 5  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
5 incom 3269 . . . . 5  |-  ( ( F  |`  A )  i^i  _I  )  =  (  _I  i^i  ( F  |`  A ) )
6 inres 4880 . . . . 5  |-  ( F  i^i  (  _I  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
74, 5, 63eqtr4i 2283 . . . 4  |-  ( ( F  |`  A )  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) )
8 fnresdm 5210 . . . . 5  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
98ineq1d 3277 . . . 4  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  _I  )  =  ( F  i^i  _I  ) )
107, 9syl5reqr 2300 . . 3  |-  ( F  Fn  A  ->  ( F  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) ) )
1110dmeqd 4788 . 2  |-  ( F  Fn  A  ->  dom  (  F  i^i  _I  )  =  dom  (  F  i^i  (  _I  |`  A ) ) )
12 fnresi 5218 . . 3  |-  (  _I  |`  A )  Fn  A
13 fndmin 5484 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  ->  dom  (  F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) } )
1412, 13mpan2 655 . 2  |-  ( F  Fn  A  ->  dom  (  F  i^i  (  _I  |`  A ) )  =  { x  e.  A  |  ( F `
 x )  =  ( (  _I  |`  A ) `
 x ) } )
15 fvresi 5563 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1615eqeq2d 2264 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
1716rabbiia 2717 . . 3  |-  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =  x }
1817a1i 12 . 2  |-  ( F  Fn  A  ->  { x  e.  A  |  ( F `  x )  =  ( (  _I  |`  A ) `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =  x }
)
1911, 14, 183eqtrd 2289 1  |-  ( F  Fn  A  ->  dom  (  F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   {crab 2512    i^i cin 3077    _I cid 4197   dom cdm 4580    |` cres 4582    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  fnelfp  25921
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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