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Theorem subfaclefac 32709
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 568 . . . . . 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 2803 . . . . 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 13432 . . . . . 6 (𝑁 ∈ ℕ0 → (1...𝑁) ∈ Fin)
4 deranglem 32699 . . . . . 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 2838 . . . 4 (𝑁 ∈ ℕ0 → {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
7 simpl 486 . . . . 5 ((𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦) → 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))
87ss2abi 3956 . . . 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 8601 . . . 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 32699 . . . . 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 13832 . . . 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 587 . . 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 260 . 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 32706 . . 3 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))
1916derangval 32700 . . . 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 2773 . 2 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓𝑦) ≠ 𝑦)}))
22 hashfac 13910 . . . 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 13798 . . . 4 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2524fveq2d 6678 . . 3 (𝑁 ∈ ℕ0 → (!‘(♯‘(1...𝑁))) = (!‘𝑁))
2623, 25eqtr2d 2774 . 2 (𝑁 ∈ ℕ0 → (!‘𝑁) = (♯‘{𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
2715, 21, 263brtr4d 5062 1 (𝑁 ∈ ℕ0 → (𝑆𝑁) ≤ (!‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  {cab 2716  wne 2934  wral 3053  wss 3843   class class class wbr 5030  cmpt 5110  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  cdom 8553  Fincfn 8555  1c1 10616  cle 10754  0cn0 11976  ...cfz 12981  !cfa 13725  chash 13782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-oadd 8135  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-dju 9403  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-n0 11977  df-xnn0 12049  df-z 12063  df-uz 12325  df-fz 12982  df-seq 13461  df-fac 13726  df-bc 13755  df-hash 13783
This theorem is referenced by: (None)
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