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Theorem subfacp1 24864
Description: A two-term recurrence for the subfactorial. This theorem allows us to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 24855, subfac1 24856. (Contributed by Mario Carneiro, 23-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
subfacp1  |-  ( N  e.  NN  ->  ( S `  ( N  +  1 ) )  =  ( N  x.  ( ( S `  N )  +  ( S `  ( N  -  1 ) ) ) ) )
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 subfacp1
Dummy variables  g 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 derang.d . 2  |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } ) )
2 subfac.n . 2  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
3 f1oeq1 5657 . . . 4  |-  ( g  =  f  ->  (
g : ( 1 ... ( N  + 
1 ) ) -1-1-onto-> ( 1 ... ( N  + 
1 ) )  <->  f :
( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) ) ) )
4 fveq2 5720 . . . . . . 7  |-  ( z  =  y  ->  (
g `  z )  =  ( g `  y ) )
5 id 20 . . . . . . 7  |-  ( z  =  y  ->  z  =  y )
64, 5neeq12d 2613 . . . . . 6  |-  ( z  =  y  ->  (
( g `  z
)  =/=  z  <->  ( g `  y )  =/=  y
) )
76cbvralv 2924 . . . . 5  |-  ( A. z  e.  ( 1 ... ( N  + 
1 ) ) ( g `  z )  =/=  z  <->  A. y  e.  ( 1 ... ( N  +  1 ) ) ( g `  y )  =/=  y
)
8 fveq1 5719 . . . . . . 7  |-  ( g  =  f  ->  (
g `  y )  =  ( f `  y ) )
98neeq1d 2611 . . . . . 6  |-  ( g  =  f  ->  (
( g `  y
)  =/=  y  <->  ( f `  y )  =/=  y
) )
109ralbidv 2717 . . . . 5  |-  ( g  =  f  ->  ( A. y  e.  (
1 ... ( N  + 
1 ) ) ( g `  y )  =/=  y  <->  A. y  e.  ( 1 ... ( N  +  1 ) ) ( f `  y )  =/=  y
) )
117, 10syl5bb 249 . . . 4  |-  ( g  =  f  ->  ( A. z  e.  (
1 ... ( N  + 
1 ) ) ( g `  z )  =/=  z  <->  A. y  e.  ( 1 ... ( N  +  1 ) ) ( f `  y )  =/=  y
) )
123, 11anbi12d 692 . . 3  |-  ( g  =  f  ->  (
( g : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  A. z  e.  ( 1 ... ( N  +  1 ) ) ( g `  z )  =/=  z
)  <->  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  A. y  e.  ( 1 ... ( N  +  1 ) ) ( f `  y )  =/=  y
) ) )
1312cbvabv 2554 . 2  |-  { g  |  ( g : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  A. z  e.  ( 1 ... ( N  +  1 ) ) ( g `  z )  =/=  z
) }  =  {
f  |  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> ( 1 ... ( N  +  1 ) )  /\  A. y  e.  ( 1 ... ( N  +  1 ) ) ( f `  y )  =/=  y
) }
141, 2, 13subfacp1lem6 24863 1  |-  ( N  e.  NN  ->  ( S `  ( N  +  1 ) )  =  ( N  x.  ( ( S `  N )  +  ( S `  ( N  -  1 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697    e. cmpt 4258   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Fincfn 7101   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   NNcn 9992   NN0cn0 10213   ...cfz 11035   #chash 11610
This theorem is referenced by:  subfacval2  24865
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-hash 11611
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