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Theorem f1omvdco3 26540
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 286 . . . . 5  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( -.  X  e.  dom  ( F 
\  _I  )  <->  -.  X  e.  dom  ( G  \  _I  ) ) )
2 disjsn 3727 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( F 
\  _I  ) )
3 disj2 3536 . . . . . . 7  |-  ( ( dom  ( F  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( F  \  _I  )  C_  ( _V  \  { X } ) )
42, 3bitr3i 242 . . . . . 6  |-  ( -.  X  e.  dom  ( F  \  _I  )  <->  dom  ( F 
\  _I  )  C_  ( _V  \  { X } ) )
5 disjsn 3727 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( G 
\  _I  ) )
6 disj2 3536 . . . . . . 7  |-  ( ( dom  ( G  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( G  \  _I  )  C_  ( _V  \  { X } ) )
75, 6bitr3i 242 . . . . . 6  |-  ( -.  X  e.  dom  ( G  \  _I  )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) )
84, 7bibi12i 306 . . . . 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 240 . . . 4  |-  ( ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G  \  _I  ) )  <->  ( dom  ( F  \  _I  )  C_  ( _V  \  { X } )  <->  dom  ( G 
\  _I  )  C_  ( _V  \  { X } ) ) )
109notbii 287 . . 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 1296 . . 3  |-  ( ( X  e.  dom  ( F  \  _I  )  \/_  X  e.  dom  ( G 
\  _I  ) )  <->  -.  ( X  e.  dom  ( F  \  _I  )  <->  X  e.  dom  ( G 
\  _I  ) ) )
12 df-xor 1296 . . 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 268 . 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 26539 . . 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 3536 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } ) )
16 disjsn 3727 . . . . 5  |-  ( ( dom  ( ( F  o.  G )  \  _I  )  i^i  { X } )  =  (/)  <->  -.  X  e.  dom  ( ( F  o.  G ) 
\  _I  ) )
1715, 16bitr3i 242 . . . 4  |-  ( dom  ( ( F  o.  G )  \  _I  )  C_  ( _V  \  { X } )  <->  -.  X  e.  dom  ( ( F  o.  G )  \  _I  ) )
1817con2bii 322 . . 3  |-  ( X  e.  dom  ( ( F  o.  G ) 
\  _I  )  <->  -.  dom  (
( F  o.  G
)  \  _I  )  C_  ( _V  \  { X } ) )
1914, 18sylibr 203 . 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 1220 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 176    /\ w3a 934    \/_ wxo 1295    = wceq 1633    e. wcel 1701   _Vcvv 2822    \ cdif 3183    i^i cin 3185    C_ wss 3186   (/)c0 3489   {csn 3674    _I cid 4341   dom cdm 4726    o. ccom 4730   -1-1-onto->wf1o 5291
This theorem is referenced by:  psgnunilem5  26565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-xor 1296  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300
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