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Theorem f1omvdco3 17090
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 297 . . . . 5  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( -.  X  e.  dom  ( F 
\  _I  )  <->  -.  X  e.  dom  ( G  \  _I  ) ) )
2 disjsn 4032 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( F 
\  _I  ) )
3 disj2 3812 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( F  \  _I  )  C_  ( _V  \  { X } ) )
42, 3bitr3i 255 . . . . . 6  |-  ( -.  X  e.  dom  ( F  \  _I  )  <->  dom  ( F 
\  _I  )  C_  ( _V  \  { X } ) )
5 disjsn 4032 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( G 
\  _I  ) )
6 disj2 3812 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) )
75, 6bitr3i 255 . . . . . 6  |-  ( -.  X  e.  dom  ( G  \  _I  )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) )
84, 7bibi12i 317 . . . . 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 253 . . . 4  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
109notbii 298 . . 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 1406 . . 3  |-  ( ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) )  <->  -.  ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G 
\  _I  ) ) )
12 df-xor 1406 . . 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 281 . 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 17089 . . 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 3812 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } ) )
16 disjsn 4032 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( ( F  o.  G ) 
\  _I  ) )
1715, 16bitr3i 255 . . . 4  |-  ( dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } )  <->  -.  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
1817con2bii 334 . . 3  |-  ( X  e.  dom  ( ( F  o.  G ) 
\  _I  )  <->  -.  dom  (
( F  o.  G
)  \  _I  )  C_  ( _V  \  { X } ) )
1914, 18sylibr 216 . 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 1306 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 188    /\ w3a 985    \/_ wxo 1405    = wceq 1444    e. wcel 1887   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968    _I cid 4744   dom cdm 4834    o. ccom 4838   -1-1-onto->wf1o 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-xor 1406  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590
This theorem is referenced by:  psgnunilem5  17135
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