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Theorem subfacval 35178
Description: The subfactorial is defined as the number of derangements (see derangval 35172) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-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
subfacval (𝑁 ∈ ℕ0 → (𝑆𝑁) = (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
StepHypRef Expression
1 oveq2 7439 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21fveq2d 6910 . 2 (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁)))
3 subfac.n . 2 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
4 fvex 6919 . 2 (𝐷‘(1...𝑁)) ∈ V
52, 3, 4fvmpt 7016 1 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  cmpt 5225  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Fincfn 8985  1c1 11156  0cn0 12526  ...cfz 13547  chash 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434
This theorem is referenced by:  derangen2  35179  subfaclefac  35181  subfac0  35182  subfac1  35183  subfacp1lem6  35190
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