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Theorem subfacval 28368
Description: The subfactorial is defined as the number of derangements (see derangval 28362) 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 6293 . . 3  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21fveq2d 5870 . 2  |-  ( n  =  N  ->  ( D `  ( 1 ... n ) )  =  ( D `  (
1 ... N ) ) )
3 subfac.n . 2  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
4 fvex 5876 . 2  |-  ( D `
 ( 1 ... N ) )  e. 
_V
52, 3, 4fvmpt 5951 1  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( D `  (
1 ... N ) ) )
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  (class class class)co 6285   Fincfn 7517   1c1 9494   NN0cn0 10796   ...cfz 11673   #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-fv 5596  df-ov 6288
This theorem is referenced by:  derangen2  28369  subfaclefac  28371  subfac0  28372  subfac1  28373  subfacp1lem6  28380
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