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Theorem subfaclefac 28260
Description: The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-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
subfaclefac  |-  ( N  e.  NN0  ->  ( S `
 N )  <_ 
( ! `  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 subfaclefac
StepHypRef Expression
1 anidm 644 . . . . . 6  |-  ( ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )  <->  f :
( 1 ... N
)
-1-1-onto-> ( 1 ... N
) )
21abbii 2601 . . . . 5  |-  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  f :
( 1 ... N
)
-1-1-onto-> ( 1 ... N
) ) }  =  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }
3 fzfid 12047 . . . . . 6  |-  ( N  e.  NN0  ->  ( 1 ... N )  e. 
Fin )
4 deranglem 28250 . . . . . 6  |-  ( ( 1 ... N )  e.  Fin  ->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  f :
( 1 ... N
)
-1-1-onto-> ( 1 ... N
) ) }  e.  Fin )
53, 4syl 16 . . . . 5  |-  ( N  e.  NN0  ->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  f :
( 1 ... N
)
-1-1-onto-> ( 1 ... N
) ) }  e.  Fin )
62, 5syl5eqelr 2560 . . . 4  |-  ( N  e.  NN0  ->  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) }  e.  Fin )
7 simpl 457 . . . . 5  |-  ( ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  A. y  e.  ( 1 ... N ) ( f `  y )  =/=  y )  -> 
f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
87ss2abi 3572 . . . 4  |-  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  C_  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) }
9 ssdomg 7558 . . . 4  |-  ( { f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) }  e.  Fin  ->  ( { f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  C_  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) }  ->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  ~<_  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
106, 8, 9mpisyl 18 . . 3  |-  ( N  e.  NN0  ->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  ~<_  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
11 deranglem 28250 . . . . 5  |-  ( ( 1 ... N )  e.  Fin  ->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  e.  Fin )
123, 11syl 16 . . . 4  |-  ( N  e.  NN0  ->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  e.  Fin )
13 hashdom 12411 . . . 4  |-  ( ( { f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  A. y  e.  ( 1 ... N ) ( f `  y )  =/=  y ) }  e.  Fin  /\  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) }  e.  Fin )  ->  ( ( # `  { f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  A. y  e.  ( 1 ... N ) ( f `  y )  =/=  y ) } )  <_  ( # `  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  <->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  ~<_  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
1412, 6, 13syl2anc 661 . . 3  |-  ( N  e.  NN0  ->  ( (
# `  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) } )  <_ 
( # `  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  <->  { f  |  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) }  ~<_  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
1510, 14mpbird 232 . 2  |-  ( N  e.  NN0  ->  ( # `  { f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  A. y  e.  ( 1 ... N ) ( f `  y )  =/=  y ) } )  <_  ( # `  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } ) )
16 derang.d . . . 4  |-  D  =  ( x  e.  Fin  |->  ( # `  { f  |  ( f : x -1-1-onto-> x  /\  A. y  e.  x  ( f `  y )  =/=  y
) } ) )
17 subfac.n . . . 4  |-  S  =  ( n  e.  NN0  |->  ( D `  ( 1 ... n ) ) )
1816, 17subfacval 28257 . . 3  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( D `  (
1 ... N ) ) )
1916derangval 28251 . . . 4  |-  ( ( 1 ... N )  e.  Fin  ->  ( D `  ( 1 ... N ) )  =  ( # `  {
f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) } ) )
203, 19syl 16 . . 3  |-  ( N  e.  NN0  ->  ( D `
 ( 1 ... N ) )  =  ( # `  {
f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) } ) )
2118, 20eqtrd 2508 . 2  |-  ( N  e.  NN0  ->  ( S `
 N )  =  ( # `  {
f  |  ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  /\  A. y  e.  ( 1 ... N
) ( f `  y )  =/=  y
) } ) )
22 hashfac 12469 . . . 4  |-  ( ( 1 ... N )  e.  Fin  ->  ( # `
 { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  =  ( ! `
 ( # `  (
1 ... N ) ) ) )
233, 22syl 16 . . 3  |-  ( N  e.  NN0  ->  ( # `  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  =  ( ! `  ( # `
 ( 1 ... N ) ) ) )
24 hashfz1 12383 . . . 4  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
2524fveq2d 5868 . . 3  |-  ( N  e.  NN0  ->  ( ! `
 ( # `  (
1 ... N ) ) )  =  ( ! `
 N ) )
2623, 25eqtr2d 2509 . 2  |-  ( N  e.  NN0  ->  ( ! `
 N )  =  ( # `  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } ) )
2715, 21, 263brtr4d 4477 1  |-  ( N  e.  NN0  ->  ( S `
 N )  <_ 
( ! `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   A.wral 2814    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282    ~<_ cdom 7511   Fincfn 7513   1c1 9489    <_ cle 9625   NN0cn0 10791   ...cfz 11668   !cfa 12317   #chash 12369
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12072  df-fac 12318  df-bc 12345  df-hash 12370
This theorem is referenced by: (None)
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