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Theorem fnnfpeq0 26753
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 3651 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  x }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  x )
2 fvresi 5927 . . . . . . 7  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
32eqeq2d 2449 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
43adantl 454 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  x )  =  x ) )
5 nne 2607 . . . . 5  |-  ( -.  ( F `  x
)  =/=  x  <->  ( F `  x )  =  x )
64, 5syl6rbbr 257 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =/=  x  <->  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
76ralbidva 2723 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  -.  ( F `  x
)  =/=  x  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
81, 7syl5bb 250 . 2  |-  ( F  Fn  A  ->  ( { x  e.  A  |  ( F `  x )  =/=  x }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
9 fndifnfp 26751 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
109eqeq1d 2446 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  { x  e.  A  |  ( F `  x )  =/=  x }  =  (/) ) )
11 fnresi 5565 . . 3  |-  (  _I  |`  A )  Fn  A
12 eqfnfv 5830 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  -> 
( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `  x
) ) )
1311, 12mpan2 654 . 2  |-  ( F  Fn  A  ->  ( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
148, 10, 133bitr4d 278 1  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711    \ cdif 3319   (/)c0 3630    _I cid 4496   dom cdm 4881    |` cres 4883    Fn wfn 5452   ` cfv 5457
This theorem is referenced by:  symggen  27402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
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