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Theorem fnnfpeq0 26861
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 3489 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  x }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  x )
2 fvresi 5727 . . . . . . 7  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
32eqeq2d 2307 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
43adantl 452 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  x )  =  x ) )
5 nne 2463 . . . . 5  |-  ( -.  ( F `  x
)  =/=  x  <->  ( F `  x )  =  x )
64, 5syl6rbbr 255 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =/=  x  <->  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
76ralbidva 2572 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  -.  ( F `  x
)  =/=  x  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
81, 7syl5bb 248 . 2  |-  ( F  Fn  A  ->  ( { x  e.  A  |  ( F `  x )  =/=  x }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
9 fndifnfp 26859 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
109eqeq1d 2304 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  { x  e.  A  |  ( F `  x )  =/=  x }  =  (/) ) )
11 fnresi 5377 . . 3  |-  (  _I  |`  A )  Fn  A
12 eqfnfv 5638 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  -> 
( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `  x
) ) )
1311, 12mpan2 652 . 2  |-  ( F  Fn  A  ->  ( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
148, 10, 133bitr4d 276 1  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    \ cdif 3162   (/)c0 3468    _I cid 4320   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  symggen  27514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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