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Theorem subfacval 24859
Description: The subfactorial is defined as the number of derangements (see derangval 24853) 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 6089 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21fveq2d 5732 . 2  |-  ( n  =  N  ->  ( D `  ( 1 ... n ) )  =  ( D `  (
1 ... N ) ) )
3 subfac.n . 2  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
4 fvex 5742 . 2  |-  ( D `
 ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5806 1  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( D `  (
1 ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705    e. cmpt 4266   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   Fincfn 7109   1c1 8991   NN0cn0 10221   ...cfz 11043   #chash 11618
This theorem is referenced by:  derangen2  24860  subfaclefac  24862  subfac0  24863  subfac1  24864  subfacp1lem6  24871
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-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084
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