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Theorem subfacval 29725
Description: The subfactorial is defined as the number of derangements (see derangval 29719) 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 6304 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21fveq2d 5876 . 2  |-  ( n  =  N  ->  ( D `  ( 1 ... n ) )  =  ( D `  (
1 ... N ) ) )
3 subfac.n . 2  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
4 fvex 5882 . 2  |-  ( D `
 ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5955 1  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( D `  (
1 ... N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   {cab 2405    =/= wne 2616   A.wral 2773    |-> cmpt 4475   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   Fincfn 7568   1c1 9529   NN0cn0 10858   ...cfz 11771   #chash 12501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299
This theorem is referenced by:  derangen2  29726  subfaclefac  29728  subfac0  29729  subfac1  29730  subfacp1lem6  29737
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