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Theorem fnnfpeq0 6093
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 3807 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  x }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  x )
2 fvresi 6088 . . . . . . 7  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
32eqeq2d 2481 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
43adantl 466 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  x )  =  x ) )
5 nne 2668 . . . . 5  |-  ( -.  ( F `  x
)  =/=  x  <->  ( F `  x )  =  x )
64, 5syl6rbbr 264 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( -.  ( F `
 x )  =/=  x  <->  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
76ralbidva 2900 . . 3  |-  ( F  Fn  A  ->  ( A. x  e.  A  -.  ( F `  x
)  =/=  x  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
81, 7syl5bb 257 . 2  |-  ( F  Fn  A  ->  ( { x  e.  A  |  ( F `  x )  =/=  x }  =  (/)  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
9 fndifnfp 6091 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
109eqeq1d 2469 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  { x  e.  A  |  ( F `  x )  =/=  x }  =  (/) ) )
11 fnresi 5698 . . 3  |-  (  _I  |`  A )  Fn  A
12 eqfnfv 5976 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  -> 
( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `  x
) ) )
1311, 12mpan2 671 . 2  |-  ( F  Fn  A  ->  ( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `
 x ) ) )
148, 10, 133bitr4d 285 1  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  F  =  (  _I  |`  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818    \ cdif 3473   (/)c0 3785    _I cid 4790   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596
This theorem is referenced by:  symggen  16310  m1detdiag  18906  mdetdiaglem  18907
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