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Theorem subfacval 34778
Description: The subfactorial is defined as the number of derangements (see derangval 34772) 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 7423 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21fveq2d 6896 . 2 (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁)))
3 subfac.n . 2 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
4 fvex 6905 . 2 (𝐷‘(1...𝑁)) ∈ V
52, 3, 4fvmpt 7000 1 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  {cab 2705  wne 2936  wral 3057  cmpt 5226  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7415  Fincfn 8958  1c1 11134  0cn0 12497  ...cfz 13511  chash 14316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7418
This theorem is referenced by:  derangen2  34779  subfaclefac  34781  subfac0  34782  subfac1  34783  subfacp1lem6  34790
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