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Theorem f1omvdmvd 16682
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 459 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  dom  ( F  \  _I  )
)
2 f1ofn 5754 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
32adantr 463 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F  Fn  A
)
4 difss 3567 . . . . . . . . 9  |-  ( F 
\  _I  )  C_  F
5 dmss 5142 . . . . . . . . 9  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
64, 5ax-mp 5 . . . . . . . 8  |-  dom  ( F  \  _I  )  C_  dom  F
7 f1odm 5757 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  dom  F  =  A )
86, 7syl5sseq 3487 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  C_  A )
98sselda 3439 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  A
)
10 fnelnfp 6035 . . . . . 6  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
113, 9, 10syl2anc 659 . . . . 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 5752 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F : A -1-1-> A )
1413adantr 463 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A -1-1-> A )
15 f1of 5753 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  F : A
--> A )
1615adantr 463 . . . . . . 7  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A --> A )
1716, 9ffvelrnd 5964 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  A
)
18 f1fveq 6105 . . . . . 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 1226 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =  ( F `  X
)  <->  ( F `  X )  =  X ) )
2019necon3bid 2659 . . . 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 6035 . . . 4  |-  ( ( F  Fn  A  /\  ( F `  X )  e.  A )  -> 
( ( F `  X )  e.  dom  ( F  \  _I  )  <->  ( F `  ( F `
 X ) )  =/=  ( F `  X ) ) )
233, 17, 22syl2anc 659 . . 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 4094 . 2  |-  ( ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } )  <->  ( ( F `  X )  e.  dom  ( F  \  _I  )  /\  ( F `  X )  =/=  X ) )
2624, 12, 25sylanbrc 662 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 367    = wceq 1403    e. wcel 1840    =/= wne 2596    \ cdif 3408    C_ wss 3411   {csn 3969    _I cid 4730   dom cdm 4940    Fn wfn 5518   -->wf 5519   -1-1->wf1 5520   -1-1-onto->wf1o 5522   ` cfv 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-f1o 5530  df-fv 5531
This theorem is referenced by:  f1otrspeq  16686  symggen  16709
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