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Theorem subfacval 23704
Description: The subfactorial is defined as the number of derangements (see derangval 23698) of the set  ( 1 ... N ). (Contributed by Mario Carneiro, 21-Jan-2015.)
Hypotheses
Ref Expression
derang.d  |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } ) )
subfac.n  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
Assertion
Ref Expression
subfacval  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( D `  (
1 ... N ) ) )
Distinct variable groups:    f, n, x, y, N    D, n    S, n, x, y
Allowed substitution hints:    D( x, y, f)    S( f)

Proof of Theorem subfacval
StepHypRef Expression
1 oveq2 5866 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21fveq2d 5529 . 2  |-  ( n  =  N  ->  ( D `  ( 1 ... n ) )  =  ( D `  (
1 ... N ) ) )
3 subfac.n . 2  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
4 fvex 5539 . 2  |-  ( D `
 ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5602 1  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( D `  (
1 ... N ) ) )
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  (class class class)co 5858   Fincfn 6863   1c1 8738   NN0cn0 9965   ...cfz 10782   #chash 11337
This theorem is referenced by:  derangen2  23705  subfaclefac  23707  subfac0  23708  subfac1  23709  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-fv 5263  df-ov 5861
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