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Theorem fninfp 26121
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 4961 . . . . . 6  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( (  _I  i^i  F )  |`  A )
2 incom 3336 . . . . . . 7  |-  (  _I 
i^i  F )  =  ( F  i^i  _I  )
32reseq1i 4939 . . . . . 6  |-  ( (  _I  i^i  F )  |`  A )  =  ( ( F  i^i  _I  )  |`  A )
41, 3eqtri 2278 . . . . 5  |-  (  _I 
i^i  ( F  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
5 incom 3336 . . . . 5  |-  ( ( F  |`  A )  i^i  _I  )  =  (  _I  i^i  ( F  |`  A ) )
6 inres 4961 . . . . 5  |-  ( F  i^i  (  _I  |`  A ) )  =  ( ( F  i^i  _I  )  |`  A )
74, 5, 63eqtr4i 2288 . . . 4  |-  ( ( F  |`  A )  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) )
8 fnresdm 5291 . . . . 5  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
98ineq1d 3344 . . . 4  |-  ( F  Fn  A  ->  (
( F  |`  A )  i^i  _I  )  =  ( F  i^i  _I  ) )
107, 9syl5reqr 2305 . . 3  |-  ( F  Fn  A  ->  ( F  i^i  _I  )  =  ( F  i^i  (  _I  |`  A ) ) )
1110dmeqd 4869 . 2  |-  ( F  Fn  A  ->  dom  (  F  i^i  _I  )  =  dom  (  F  i^i  (  _I  |`  A ) ) )
12 fnresi 5299 . . 3  |-  (  _I  |`  A )  Fn  A
13 fndmin 5566 . . 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 5645 . . . . 5  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
1615eqeq2d 2269 . . . 4  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
1716rabbiia 2753 . . 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 2294 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 2522    i^i cin 3126    _I cid 4276   dom cdm 4661    |` cres 4663    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  fnelfp  26122
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689
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