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Theorem f1omvdco3 16347
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 295 . . . . 5  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( -.  X  e.  dom  ( F 
\  _I  )  <->  -.  X  e.  dom  ( G  \  _I  ) ) )
2 disjsn 4094 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( F 
\  _I  ) )
3 disj2 3879 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( F  \  _I  )  C_  ( _V  \  { X } ) )
42, 3bitr3i 251 . . . . . 6  |-  ( -.  X  e.  dom  ( F  \  _I  )  <->  dom  ( F 
\  _I  )  C_  ( _V  \  { X } ) )
5 disjsn 4094 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( G 
\  _I  ) )
6 disj2 3879 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) )
75, 6bitr3i 251 . . . . . 6  |-  ( -.  X  e.  dom  ( G  \  _I  )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) )
84, 7bibi12i 315 . . . . 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 249 . . . 4  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
109notbii 296 . . 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 1361 . . 3  |-  ( ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) )  <->  -.  ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G 
\  _I  ) ) )
12 df-xor 1361 . . 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 277 . 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 16346 . . 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 3879 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } ) )
16 disjsn 4094 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( ( F  o.  G ) 
\  _I  ) )
1715, 16bitr3i 251 . . . 4  |-  ( dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } )  <->  -.  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
1817con2bii 332 . . 3  |-  ( X  e.  dom  ( ( F  o.  G ) 
\  _I  )  <->  -.  dom  (
( F  o.  G
)  \  _I  )  C_  ( _V  \  { X } ) )
1914, 18sylibr 212 . 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 1266 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 184    /\ w3a 973    \/_ wxo 1360    = wceq 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033    _I cid 4796   dom cdm 5005    o. ccom 5009   -1-1-onto->wf1o 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  psgnunilem5  16392
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