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Theorem f1omvdmvd 27363
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 448 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  dom  ( F  \  _I  )
)
2 f1ofn 5675 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F  Fn  A )
32adantr 452 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F  Fn  A
)
4 difss 3474 . . . . . . . . 9  |-  ( F 
\  _I  )  C_  F
5 dmss 5069 . . . . . . . . 9  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
64, 5ax-mp 8 . . . . . . . 8  |-  dom  ( F  \  _I  )  C_  dom  F
7 f1odm 5678 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  dom  F  =  A )
86, 7syl5sseq 3396 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  C_  A )
98sselda 3348 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  A
)
10 fnelnfp 26738 . . . . . 6  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( X  e.  dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X ) )
113, 9, 10syl2anc 643 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( X  e. 
dom  ( F  \  _I  )  <->  ( F `  X )  =/=  X
) )
121, 11mpbid 202 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  =/=  X
)
13 f1of1 5673 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  F : A -1-1-> A )
1413adantr 452 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A -1-1-> A )
15 f1of 5674 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  F : A
--> A )
1615adantr 452 . . . . . . 7  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  F : A --> A )
1716, 9ffvelrnd 5871 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  A
)
18 f1fveq 6008 . . . . . 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 1182 . . . . 5  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =  ( F `  X
)  <->  ( F `  X )  =  X ) )
2019necon3bid 2636 . . . 4  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( ( F `
 ( F `  X ) )  =/=  ( F `  X
)  <->  ( F `  X )  =/=  X
) )
2112, 20mpbird 224 . . 3  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  ( F `  X ) )  =/=  ( F `
 X ) )
22 fnelnfp 26738 . . . 4  |-  ( ( F  Fn  A  /\  ( F `  X )  e.  A )  -> 
( ( F `  X )  e.  dom  ( F  \  _I  )  <->  ( F `  ( F `
 X ) )  =/=  ( F `  X ) ) )
233, 17, 22syl2anc 643 . . 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 224 . 2  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  ( F `  X )  e.  dom  ( F  \  _I  )
)
25 eldifsn 3927 . 2  |-  ( ( F `  X )  e.  ( dom  ( F  \  _I  )  \  { X } )  <->  ( ( F `  X )  e.  dom  ( F  \  _I  )  /\  ( F `  X )  =/=  X ) )
2624, 12, 25sylanbrc 646 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    C_ wss 3320   {csn 3814    _I cid 4493   dom cdm 4878    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454
This theorem is referenced by:  f1otrspeq  27367  symggen  27388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-f1o 5461  df-fv 5462
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