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Theorem f1omvdco3 17168
Description: If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Assertion
Ref Expression
f1omvdco3  |-  ( ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( X  e.  dom  ( F 
\  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) ) )  ->  X  e.  dom  ( ( F  o.  G )  \  _I  ) )

Proof of Theorem f1omvdco3
StepHypRef Expression
1 notbi 302 . . . . 5  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( -.  X  e.  dom  ( F 
\  _I  )  <->  -.  X  e.  dom  ( G  \  _I  ) ) )
2 disjsn 4023 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( F 
\  _I  ) )
3 disj2 3816 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( F  \  _I  )  C_  ( _V  \  { X } ) )
42, 3bitr3i 259 . . . . . 6  |-  ( -.  X  e.  dom  ( F  \  _I  )  <->  dom  ( F 
\  _I  )  C_  ( _V  \  { X } ) )
5 disjsn 4023 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( G 
\  _I  ) )
6 disj2 3816 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) )
75, 6bitr3i 259 . . . . . 6  |-  ( -.  X  e.  dom  ( G  \  _I  )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) )
84, 7bibi12i 322 . . . . 5  |-  ( ( -.  X  e.  dom  ( F  \  _I  )  <->  -.  X  e.  dom  ( G  \  _I  ) )  <-> 
( dom  ( F  \  _I  )  C_  ( _V  \  { X }
)  <->  dom  ( G  \  _I  )  C_  ( _V 
\  { X }
) ) )
91, 8bitri 257 . . . 4  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
109notbii 303 . . 3  |-  ( -.  ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G 
\  _I  ) )  <->  -.  ( dom  ( F 
\  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
11 df-xor 1431 . . 3  |-  ( ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) )  <->  -.  ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G 
\  _I  ) ) )
12 df-xor 1431 . . 3  |-  ( ( dom  ( F  \  _I  )  C_  ( _V 
\  { X }
)  \/_  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) )  <->  -.  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
1310, 11, 123bitr4i 285 . 2  |-  ( ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) )  <-> 
( dom  ( F  \  _I  )  C_  ( _V  \  { X }
)  \/_  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
14 f1omvdco2 17167 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  \/_  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) ) )  ->  -.  dom  (
( F  o.  G
)  \  _I  )  C_  ( _V  \  { X } ) )
15 disj2 3816 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } ) )
16 disjsn 4023 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( ( F  o.  G ) 
\  _I  ) )
1715, 16bitr3i 259 . . . 4  |-  ( dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } )  <->  -.  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
1817con2bii 339 . . 3  |-  ( X  e.  dom  ( ( F  o.  G ) 
\  _I  )  <->  -.  dom  (
( F  o.  G
)  \  _I  )  C_  ( _V  \  { X } ) )
1914, 18sylibr 217 . 2  |-  ( ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  \/_  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) ) )  ->  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
2013, 19syl3an3b 1330 1  |-  ( ( F : A -1-1-onto-> A  /\  G : A -1-1-onto-> A  /\  ( X  e.  dom  ( F 
\  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) ) )  ->  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ w3a 1007    \/_ wxo 1430    = wceq 1452    e. wcel 1904   _Vcvv 3031    \ cdif 3387    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959    _I cid 4749   dom cdm 4839    o. ccom 4843   -1-1-onto->wf1o 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-xor 1431  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597
This theorem is referenced by:  psgnunilem5  17213
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