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Theorem derangval 23698
Description: Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypothesis
Ref Expression
derang.d  |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } ) )
Assertion
Ref Expression
derangval  |-  ( A  e.  Fin  ->  ( D `  A )  =  ( # `  {
f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y
) } ) )
Distinct variable group:    x, f, y, A
Allowed substitution hints:    D( x, y, f)

Proof of Theorem derangval
StepHypRef Expression
1 f1oeq2 5464 . . . . . 6  |-  ( x  =  A  ->  (
f : x -1-1-onto-> x  <->  f : A
-1-1-onto-> x ) )
2 f1oeq3 5465 . . . . . 6  |-  ( x  =  A  ->  (
f : A -1-1-onto-> x  <->  f : A
-1-1-onto-> A ) )
31, 2bitrd 244 . . . . 5  |-  ( x  =  A  ->  (
f : x -1-1-onto-> x  <->  f : A
-1-1-onto-> A ) )
4 raleq 2736 . . . . 5  |-  ( x  =  A  ->  ( A. y  e.  x  ( f `  y
)  =/=  y  <->  A. y  e.  A  ( f `  y )  =/=  y
) )
53, 4anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
)  <->  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y
) ) )
65abbidv 2397 . . 3  |-  ( x  =  A  ->  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) }  =  {
f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y
) } )
76fveq2d 5529 . 2  |-  ( x  =  A  ->  ( # `
 { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } )  =  ( # `  {
f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y
) } ) )
8 derang.d . 2  |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } ) )
9 fvex 5539 . 2  |-  ( # `  { f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y ) } )  e.  _V
107, 8, 9fvmpt 5602 1  |-  ( A  e.  Fin  ->  ( D `  A )  =  ( # `  {
f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y
) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543    e. cmpt 4077   -1-1-onto->wf1o 5254   ` cfv 5255   Fincfn 6863   #chash 11337
This theorem is referenced by:  derang0  23700  derangsn  23701  derangenlem  23702  subfaclefac  23707  subfacp1lem3  23713  subfacp1lem5  23715  subfacp1lem6  23716
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-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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