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Theorem f1omvdcnv 27490
Description: A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Assertion
Ref Expression
f1omvdcnv  |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  dom  ( F  \  _I  ) )

Proof of Theorem f1omvdcnv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 f1ocnvfvb 5811 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A  /\  x  e.  A )  ->  ( ( F `  x )  =  x  <-> 
( `' F `  x )  =  x ) )
213anidm23 1241 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( F `  x )  =  x  <-> 
( `' F `  x )  =  x ) )
32bicomd 192 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( `' F `  x )  =  x  <-> 
( F `  x
)  =  x ) )
43necon3bid 2494 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( `' F `  x )  =/=  x  <->  ( F `  x )  =/=  x ) )
54rabbidva 2792 . 2  |-  ( F : A -1-1-onto-> A  ->  { x  e.  A  |  ( `' F `  x )  =/=  x }  =  { x  e.  A  |  ( F `  x )  =/=  x } )
6 f1ocnv 5501 . . 3  |-  ( F : A -1-1-onto-> A  ->  `' F : A -1-1-onto-> A )
7 f1ofn 5489 . . 3  |-  ( `' F : A -1-1-onto-> A  ->  `' F  Fn  A
)
8 fndifnfp 26859 . . 3  |-  ( `' F  Fn  A  ->  dom  ( `' F  \  _I  )  =  {
x  e.  A  | 
( `' F `  x )  =/=  x } )
96, 7, 83syl 18 . 2  |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  { x  e.  A  |  ( `' F `  x )  =/=  x } )
10 f1ofn 5489 . . 3  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
11 fndifnfp 26859 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
1210, 11syl 15 . 2  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
135, 9, 123eqtr4d 2338 1  |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  dom  ( F  \  _I  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162    _I cid 4320   `'ccnv 4704   dom cdm 4705    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271
This theorem is referenced by:  f1omvdco2  27494  symgsssg  27511  symgfisg  27512
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-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-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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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