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Theorem f1omvdmvd 27386
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 447 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  dom  ( F  \  _I  )
)
2 f1ofn 5473 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
32adantr 451 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F  Fn  A
)
4 difss 3303 . . . . . . . . 9  |-  ( F 
\  _I  )  C_  F
5 dmss 4878 . . . . . . . . 9  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
64, 5ax-mp 8 . . . . . . . 8  |-  dom  ( F  \  _I  )  C_  dom  F
7 f1odm 5476 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  dom  F  =  A )
86, 7syl5sseq 3226 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  C_  A )
98sselda 3180 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  A
)
10 fnelnfp 26757 . . . . . 6  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
113, 9, 10syl2anc 642 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( X  e. 
dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X
) )
121, 11mpbid 201 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  =/=  X
)
13 f1of1 5471 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F : A -1-1-> A )
1413adantr 451 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A -1-1-> A )
15 f1of 5472 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  F : A
--> A )
1615adantr 451 . . . . . . 7  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A --> A )
17 ffvelrn 5663 . . . . . . 7  |-  ( ( F : A --> A  /\  X  e.  A )  ->  ( F `  X
)  e.  A )
1816, 9, 17syl2anc 642 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  A
)
19 f1fveq 5786 . . . . . 6  |-  ( ( F : A -1-1-> A  /\  ( ( F `  X )  e.  A  /\  X  e.  A
) )  ->  (
( F `  ( F `  X )
)  =  ( F `
 X )  <->  ( F `  X )  =  X ) )
2014, 18, 9, 19syl12anc 1180 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =  ( F `  X
)  <->  ( F `  X )  =  X ) )
2120necon3bid 2481 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =/=  ( F `  X
)  <->  ( F `  X )  =/=  X
) )
2212, 21mpbird 223 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  ( F `  X ) )  =/=  ( F `
 X ) )
23 fnelnfp 26757 . . . 4  |-  ( ( F  Fn  A  /\  ( F `  X )  e.  A )  -> 
( ( F `  X )  e.  dom  ( F  \  _I  )  <->  ( F `  ( F `
 X ) )  =/=  ( F `  X ) ) )
243, 18, 23syl2anc 642 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 X )  e. 
dom  ( F  \  _I  )  <->  ( F `  ( F `  X ) )  =/=  ( F `
 X ) ) )
2522, 24mpbird 223 . 2  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  dom  ( F  \  _I  )
)
26 eldifsn 3749 . 2  |-  ( ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } )  <->  ( ( F `  X )  e.  dom  ( F  \  _I  )  /\  ( F `  X )  =/=  X ) )
2725, 12, 26sylanbrc 645 1  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   {csn 3640    _I cid 4304   dom cdm 4689    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255
This theorem is referenced by:  f1otrspeq  27390  symggen  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-f1o 5262  df-fv 5263
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