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Theorem fnnfpeq0 6018
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 3747 . . 3  |-  ( { x  e.  A  | 
( F `  x
)  =/=  x }  =  (/)  <->  A. x  e.  A  -.  ( F `  x
)  =/=  x )
2 fvresi 6013 . . . . . . 7  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
32eqeq2d 2406 . . . . . 6  |-  ( x  e.  A  ->  (
( F `  x
)  =  ( (  _I  |`  A ) `  x )  <->  ( F `  x )  =  x ) )
43adantl 464 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  ( (  _I  |`  A ) `
 x )  <->  ( F `  x )  =  x ) )
5 nne 2593 . . . . 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 2828 . . 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 6016 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
109eqeq1d 2394 . 2  |-  ( F  Fn  A  ->  ( dom  ( F  \  _I  )  =  (/)  <->  { x  e.  A  |  ( F `  x )  =/=  x }  =  (/) ) )
11 fnresi 5619 . . 3  |-  (  _I  |`  A )  Fn  A
12 eqfnfv 5896 . . 3  |-  ( ( F  Fn  A  /\  (  _I  |`  A )  Fn  A )  -> 
( F  =  (  _I  |`  A )  <->  A. x  e.  A  ( F `  x )  =  ( (  _I  |`  A ) `  x
) ) )
1311, 12mpan2 669 . 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 367    = wceq 1399    e. wcel 1836    =/= wne 2587   A.wral 2742   {crab 2746    \ cdif 3399   (/)c0 3724    _I cid 4717   dom cdm 4926    |` cres 4928    Fn wfn 5504   ` cfv 5509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-sn 3958  df-pr 3960  df-op 3964  df-uni 4177  df-br 4381  df-opab 4439  df-mpt 4440  df-id 4722  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-fv 5517
This theorem is referenced by:  symggen  16631  m1detdiag  19203  mdetdiaglem  19204
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