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Theorem fnnfpeq0 26430
Description: A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
Assertion
Ref Expression
fnnfpeq0  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )

Proof of Theorem fnnfpeq0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 rabeq0 3592 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  x }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  x )
2 fvresi 5863 . . . . . . 7  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
32eqeq2d 2398 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
43adantl 453 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  x )  =  x ) )
5 nne 2554 . . . . 5  |-  ( -.  ( F `  x
)  =/=  x  <->  ( F `  x )  =  x )
64, 5syl6rbbr 256 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =/=  x  <->  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
76ralbidva 2665 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  -.  ( F `  x
)  =/=  x  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
81, 7syl5bb 249 . 2  |-  ( F  Fn  A  ->  ( { x  e.  A  |  ( F `  x )  =/=  x }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
9 fndifnfp 26428 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
109eqeq1d 2395 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  { x  e.  A  |  ( F `  x )  =/=  x }  =  (/) ) )
11 fnresi 5502 . . 3  |-  (  _I  |`  A )  Fn  A
12 eqfnfv 5766 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  -> 
( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `  x
) ) )
1311, 12mpan2 653 . 2  |-  ( F  Fn  A  ->  ( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
148, 10, 133bitr4d 277 1  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649   {crab 2653    \ cdif 3260   (/)c0 3571    _I cid 4434   dom cdm 4818    |` cres 4820    Fn wfn 5389   ` cfv 5394
This theorem is referenced by:  symggen  27080
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402
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