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Theorem subfaclefac 35182
Description: The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypotheses
Ref Expression
derang.d 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
subfac.n 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
Assertion
Ref Expression
subfaclefac (𝑁 ∈ ℕ0 → (𝑆𝑁) ≤ (!‘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 564 . . . . . 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 2808 . . . . 5 {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} = {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}
3 fzfid 14015 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
4 deranglem 35172 . . . . . 6 ((1...𝑁) ∈ Fin → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} ∈ Fin)
53, 4syl 17 . . . . 5 (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} ∈ Fin)
62, 5eqeltrrid 2845 . . . 4 (𝑁 ∈ ℕ0 → {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
7 simpl 482 . . . . 5 ((𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦) → 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))
87ss2abi 4066 . . . 4 {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ⊆ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}
9 ssdomg 9041 . . . 4 ({𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin → ({𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ⊆ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ≼ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
106, 8, 9mpisyl 21 . . 3 (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ≼ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
11 deranglem 35172 . . . . 5 ((1...𝑁) ∈ Fin → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ∈ Fin)
123, 11syl 17 . . . 4 (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ∈ Fin)
13 hashdom 14419 . . . 4 (({𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ∈ Fin ∧ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → ((♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ≼ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
1412, 6, 13syl2anc 584 . . 3 (𝑁 ∈ ℕ0 → ((♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)} ≼ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
1510, 14mpbird 257 . 2 (𝑁 ∈ ℕ0 → (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
16 derang.d . . . 4 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
17 subfac.n . . . 4 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
1816, 17subfacval 35179 . . 3 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))
1916derangval 35173 . . . 4 ((1...𝑁) ∈ Fin → (𝐷‘(1...𝑁)) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}))
203, 19syl 17 . . 3 (𝑁 ∈ ℕ0 → (𝐷‘(1...𝑁)) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}))
2118, 20eqtrd 2776 . 2 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}))
22 hashfac 14498 . . . 4 ((1...𝑁) ∈ Fin → (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
233, 22syl 17 . . 3 (𝑁 ∈ ℕ0 → (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
24 hashfz1 14386 . . . 4 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2524fveq2d 6909 . . 3 (𝑁 ∈ ℕ0 → (!‘(♯‘(1...𝑁))) = (!‘𝑁))
2623, 25eqtr2d 2777 . 2 (𝑁 ∈ ℕ0 → (!‘𝑁) = (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
2715, 21, 263brtr4d 5174 1 (𝑁 ∈ ℕ0 → (𝑆𝑁) ≤ (!‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wne 2939  wral 3060  wss 3950   class class class wbr 5142  cmpt 5224  1-1-ontowf1o 6559  cfv 6560  (class class class)co 7432  cdom 8984  Fincfn 8986  1c1 11157  cle 11297  0cn0 12528  ...cfz 13548  !cfa 14313  chash 14370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12880  df-fz 13549  df-seq 14044  df-fac 14314  df-bc 14343  df-hash 14371
This theorem is referenced by: (None)
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