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Theorem f1omvdmvd 15954
Description: A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Assertion
Ref Expression
f1omvdmvd  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } ) )

Proof of Theorem f1omvdmvd
StepHypRef Expression
1 simpr 461 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  dom  ( F  \  _I  )
)
2 f1ofn 5647 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
32adantr 465 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F  Fn  A
)
4 difss 3488 . . . . . . . . 9  |-  ( F 
\  _I  )  C_  F
5 dmss 5044 . . . . . . . . 9  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
64, 5ax-mp 5 . . . . . . . 8  |-  dom  ( F  \  _I  )  C_  dom  F
7 f1odm 5650 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  dom  F  =  A )
86, 7syl5sseq 3409 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  C_  A )
98sselda 3361 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  A
)
10 fnelnfp 5913 . . . . . 6  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
113, 9, 10syl2anc 661 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( X  e. 
dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X
) )
121, 11mpbid 210 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  =/=  X
)
13 f1of1 5645 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F : A -1-1-> A )
1413adantr 465 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A -1-1-> A )
15 f1of 5646 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  F : A
--> A )
1615adantr 465 . . . . . . 7  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A --> A )
1716, 9ffvelrnd 5849 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  A
)
18 f1fveq 5980 . . . . . 6  |-  ( ( F : A -1-1-> A  /\  ( ( F `  X )  e.  A  /\  X  e.  A
) )  ->  (
( F `  ( F `  X )
)  =  ( F `
 X )  <->  ( F `  X )  =  X ) )
1914, 17, 9, 18syl12anc 1216 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =  ( F `  X
)  <->  ( F `  X )  =  X ) )
2019necon3bid 2648 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =/=  ( F `  X
)  <->  ( F `  X )  =/=  X
) )
2112, 20mpbird 232 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  ( F `  X ) )  =/=  ( F `
 X ) )
22 fnelnfp 5913 . . . 4  |-  ( ( F  Fn  A  /\  ( F `  X )  e.  A )  -> 
( ( F `  X )  e.  dom  ( F  \  _I  )  <->  ( F `  ( F `
 X ) )  =/=  ( F `  X ) ) )
233, 17, 22syl2anc 661 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 X )  e. 
dom  ( F  \  _I  )  <->  ( F `  ( F `  X ) )  =/=  ( F `
 X ) ) )
2421, 23mpbird 232 . 2  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  dom  ( F  \  _I  )
)
25 eldifsn 4005 . 2  |-  ( ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } )  <->  ( ( F `  X )  e.  dom  ( F  \  _I  )  /\  ( F `  X )  =/=  X ) )
2624, 12, 25sylanbrc 664 1  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330    C_ wss 3333   {csn 3882    _I cid 4636   dom cdm 4845    Fn wfn 5418   -->wf 5419   -1-1->wf1 5420   -1-1-onto->wf1o 5422   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-f1o 5430  df-fv 5431
This theorem is referenced by:  f1otrspeq  15958  symggen  15981
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