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:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
subfac.n	⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))

Assertion

Ref	Expression
subfaclefac	⊢ (𝑁 ∈ ℕ0 → (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

Step	Hyp	Ref	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))
2	1	abbii 2803	. . . . 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 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)
5	3, 4	syl 17	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))} ∈ Fin)
6	2, 5	eqeltrrid 2838	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
7		simpl 486	. . . . 5 ⊢ ((𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦) → 𝑓:(1...𝑁)–1-1-onto→(1...𝑁))
8	7	ss2abi 3956	. . . 4 ⊢ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}
9		ssdomg 8601	. . . 4 ⊢ ({𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin → ({𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
10	6, 8, 9	mpisyl 21	. . 3 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(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)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
12	3, 11	syl 17	. . . 4 ⊢ (𝑁 ∈ ℕ0 → {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin)
13		hashdom 13832	. . . 4 ⊢ (({𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → ((♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
14	12, 6, 13	syl2anc 587	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ {𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)} ≼ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
15	10, 14	mpbird 260	. 2 ⊢ (𝑁 ∈ ℕ0 → (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}) ≤ (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
16		derang.d	. . . 4 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
17		subfac.n	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
18	16, 17	subfacval 32706	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁)))
19	16	derangval 32700	. . . 4 ⊢ ((1...𝑁) ∈ Fin → (𝐷‘(1...𝑁)) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
20	3, 19	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝐷‘(1...𝑁)) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
21	18, 20	eqtrd 2773	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (♯‘{𝑓 ∣ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ∧ ∀𝑦 ∈ (1...𝑁)(𝑓‘𝑦) ≠ 𝑦)}))
22		hashfac 13910	. . . 4 ⊢ ((1...𝑁) ∈ Fin → (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
23	3, 22	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ0 → (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) = (!‘(♯‘(1...𝑁))))
24		hashfz1 13798	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
25	24	fveq2d 6678	. . 3 ⊢ (𝑁 ∈ ℕ0 → (!‘(♯‘(1...𝑁))) = (!‘𝑁))
26	23, 25	eqtr2d 2774	. 2 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) = (♯‘{𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
27	15, 21, 26	3brtr4d 5062	1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) ≤ (!‘𝑁))

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-onto→wf1o 6338  ‘cfv 6339  (class class class)co 7170   ≼ cdom 8553  Fincfn 8555  1c1 10616   ≤ cle 10754  ℕ₀cn0 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)