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Theorem f1omvdco3 27369
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 287 . . . . 5  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( -.  X  e.  dom  ( F 
\  _I  )  <->  -.  X  e.  dom  ( G  \  _I  ) ) )
2 disjsn 3868 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( F 
\  _I  ) )
3 disj2 3675 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( F  \  _I  )  C_  ( _V  \  { X } ) )
42, 3bitr3i 243 . . . . . 6  |-  ( -.  X  e.  dom  ( F  \  _I  )  <->  dom  ( F 
\  _I  )  C_  ( _V  \  { X } ) )
5 disjsn 3868 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( G 
\  _I  ) )
6 disj2 3675 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) )
75, 6bitr3i 243 . . . . . 6  |-  ( -.  X  e.  dom  ( G  \  _I  )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) )
84, 7bibi12i 307 . . . . 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 241 . . . 4  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
109notbii 288 . . 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 1314 . . 3  |-  ( ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) )  <->  -.  ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G 
\  _I  ) ) )
12 df-xor 1314 . . 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 269 . 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 27368 . . 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 3675 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } ) )
16 disjsn 3868 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( ( F  o.  G ) 
\  _I  ) )
1715, 16bitr3i 243 . . . 4  |-  ( dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } )  <->  -.  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
1817con2bii 323 . . 3  |-  ( X  e.  dom  ( ( F  o.  G ) 
\  _I  )  <->  -.  dom  (
( F  o.  G
)  \  _I  )  C_  ( _V  \  { X } ) )
1914, 18sylibr 204 . 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 1222 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ w3a 936    \/_ wxo 1313    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814    _I cid 4493   dom cdm 4878    o. ccom 4882   -1-1-onto->wf1o 5453
This theorem is referenced by:  psgnunilem5  27394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-xor 1314  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
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