Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  f1omvdmvd Unicode version

Theorem f1omvdmvd 27489
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 5489 . . . . . . 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 3316 . . . . . . . . 9  |-  ( F 
\  _I  )  C_  F
5 dmss 4894 . . . . . . . . 9  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
64, 5ax-mp 8 . . . . . . . 8  |-  dom  ( F  \  _I  )  C_  dom  F
7 f1odm 5492 . . . . . . . 8  |-  ( F : A -1-1-onto-> A  ->  dom  F  =  A )
86, 7syl5sseq 3239 . . . . . . 7  |-  ( F : A -1-1-onto-> A  ->  dom  ( F 
\  _I  )  C_  A )
98sselda 3193 . . . . . 6  |-  ( ( F : A -1-1-onto-> A  /\  X  e.  dom  ( F 
\  _I  ) )  ->  X  e.  A
)
10 fnelnfp 26860 . . . . . 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 5487 . . . . . . 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 5488 . . . . . . . 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 5679 . . . . . . 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 5802 . . . . . 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 2494 . . . 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 26860 . . . 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 3762 . 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   {csn 3653    _I cid 4320   dom cdm 4705    Fn wfn 5266   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271
This theorem is referenced by:  f1otrspeq  27493  symggen  27514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279
  Copyright terms: Public domain W3C validator