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Theorem f1omvdcnv 16447
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 6170 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A  /\  x  e.  A )  ->  ( ( F `  x )  =  x  <-> 
( `' F `  x )  =  x ) )
213anidm23 1288 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( F `  x )  =  x  <-> 
( `' F `  x )  =  x ) )
32bicomd 201 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( `' F `  x )  =  x  <-> 
( F `  x
)  =  x ) )
43necon3bid 2701 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( `' F `  x )  =/=  x  <->  ( F `  x )  =/=  x ) )
54rabbidva 3086 . 2  |-  ( F : A -1-1-onto-> A  ->  { x  e.  A  |  ( `' F `  x )  =/=  x }  =  { x  e.  A  |  ( F `  x )  =/=  x } )
6 f1ocnv 5818 . . 3  |-  ( F : A -1-1-onto-> A  ->  `' F : A -1-1-onto-> A )
7 f1ofn 5807 . . 3  |-  ( `' F : A -1-1-onto-> A  ->  `' F  Fn  A
)
8 fndifnfp 6085 . . 3  |-  ( `' F  Fn  A  ->  dom  ( `' F  \  _I  )  =  {
x  e.  A  | 
( `' F `  x )  =/=  x } )
96, 7, 83syl 20 . 2  |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  { x  e.  A  |  ( `' F `  x )  =/=  x } )
10 f1ofn 5807 . . 3  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
11 fndifnfp 6085 . . 3  |-  ( F  Fn  A  ->  dom  ( F  \  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
1210, 11syl 16 . 2  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  =  { x  e.  A  |  ( F `  x )  =/=  x } )
135, 9, 123eqtr4d 2494 1  |-  ( F : A -1-1-onto-> A  ->  dom  ( `' F  \  _I  )  =  dom  ( F  \  _I  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   {crab 2797    \ cdif 3458    _I cid 4780   `'ccnv 4988   dom cdm 4989    Fn wfn 5573   -1-1-onto->wf1o 5577   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586
This theorem is referenced by:  f1omvdco2  16451  symgsssg  16470  symgfisg  16471
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