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Theorem fnelfp 6051
Description: Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnelfp  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  ( F `  X )  =  X ) )

Proof of Theorem fnelfp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fninfp 6050 . . 3  |-  ( F  Fn  A  ->  dom  ( F  i^i  _I  )  =  { x  e.  A  |  ( F `  x )  =  x } )
21eleq2d 2491 . 2  |-  ( F  Fn  A  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  X  e.  { x  e.  A  | 
( F `  x
)  =  x }
) )
3 fveq2 5825 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
4 id 22 . . . 4  |-  ( x  =  X  ->  x  =  X )
53, 4eqeq12d 2443 . . 3  |-  ( x  =  X  ->  (
( F `  x
)  =  x  <->  ( F `  X )  =  X ) )
65elrab3 3172 . 2  |-  ( X  e.  A  ->  ( X  e.  { x  e.  A  |  ( F `  x )  =  x }  <->  ( F `  X )  =  X ) )
72, 6sylan9bb 704 1  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  i^i  _I  )  <->  ( F `  X )  =  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2718    i^i cin 3378    _I cid 4706   dom cdm 4796    Fn wfn 5539   ` cfv 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-res 4808  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552
This theorem is referenced by:  ismrcd1  35452  ismrcd2  35453  istopclsd  35454
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