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Theorem derangval 28362
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 5808 . . . . . 6  |-  ( x  =  A  ->  (
f : x -1-1-onto-> x  <->  f : A
-1-1-onto-> x ) )
2 f1oeq3 5809 . . . . . 6  |-  ( x  =  A  ->  (
f : A -1-1-onto-> x  <->  f : A
-1-1-onto-> A ) )
31, 2bitrd 253 . . . . 5  |-  ( x  =  A  ->  (
f : x -1-1-onto-> x  <->  f : A
-1-1-onto-> A ) )
4 raleq 3058 . . . . 5  |-  ( x  =  A  ->  ( A. y  e.  x  ( f `  y
)  =/=  y  <->  A. y  e.  A  ( f `  y )  =/=  y
) )
53, 4anbi12d 710 . . . 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 2603 . . 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 5870 . 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 5876 . 2  |-  ( # `  { f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y ) } )  e.  _V
107, 8, 9fvmpt 5951 1  |-  ( A  e.  Fin  ->  ( D `  A )  =  ( # `  {
f  |  ( f : A -1-1-onto-> A  /\  A. y  e.  A  ( f `  y )  =/=  y
) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814    |-> cmpt 4505   -1-1-onto->wf1o 5587   ` cfv 5588   Fincfn 7517   #chash 12374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by:  derang0  28364  derangsn  28365  derangenlem  28366  subfaclefac  28371  subfacp1lem3  28377  subfacp1lem5  28379  subfacp1lem6  28380
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