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Theorem f1omvdcnv 16258
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 6164 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A  /\  x  e.  A )  ->  ( ( F `  x )  =  x  <-> 
( `' F `  x )  =  x ) )
213anidm23 1282 . . . . 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 2718 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  x  e.  A )  ->  ( ( `' F `  x )  =/=  x  <->  ( F `  x )  =/=  x ) )
54rabbidva 3097 . 2  |-  ( F : A -1-1-onto-> A  ->  { x  e.  A  |  ( `' F `  x )  =/=  x }  =  { x  e.  A  |  ( F `  x )  =/=  x } )
6 f1ocnv 5819 . . 3  |-  ( F : A -1-1-onto-> A  ->  `' F : A -1-1-onto-> A )
7 f1ofn 5808 . . 3  |-  ( `' F : A -1-1-onto-> A  ->  `' F  Fn  A
)
8 fndifnfp 6081 . . 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 5808 . . 3  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
11 fndifnfp 6081 . . 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 2511 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 1374    e. wcel 1762    =/= wne 2655   {crab 2811    \ cdif 3466    _I cid 4783   `'ccnv 4991   dom cdm 4992    Fn wfn 5574   -1-1-onto->wf1o 5578   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587
This theorem is referenced by:  f1omvdco2  16262  symgsssg  16281  symgfisg  16282
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