Theorem subfacval 35531

Description: The subfactorial is defined as the number of derangements (see derangval 35525) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-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
subfacval	⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...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 subfacval

Step	Hyp	Ref	Expression
1		oveq2 7408	. . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
2	1	fveq2d 6875	. 2 ⊢ (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁)))
3		subfac.n	. 2 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
4		fvex 6884	. 2 ⊢ (𝐷‘(1...𝑁)) ∈ V
5	2, 3, 4	fvmpt 6979	1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∀wral 3079   ↦ cmpt 5185  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400  Fincfn 8931  1c1 11089  ℕ₀cn0 12492  ...cfz 13523  ♯chash 14354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403

This theorem is referenced by:  derangen2  35532  subfaclefac  35534  subfac0  35535  subfac1  35536  subfacp1lem6  35543