Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  subfacval Structured version   Visualization version   GIF version

Theorem subfacval 32317
Description: The subfactorial is defined as the number of derangements (see derangval 32311) 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 7153 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
21fveq2d 6667 . 2 (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁)))
3 subfac.n . 2 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛)))
4 fvex 6676 . 2 (𝐷‘(1...𝑁)) ∈ V
52, 3, 4fvmpt 6761 1 (𝑁 ∈ ℕ0 → (𝑆𝑁) = (𝐷‘(1...𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {cab 2796  wne 3013  wral 3135  cmpt 5137  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Fincfn 8497  1c1 10526  0cn0 11885  ...cfz 12880  chash 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148
This theorem is referenced by:  derangen2  32318  subfaclefac  32320  subfac0  32321  subfac1  32322  subfacp1lem6  32329
  Copyright terms: Public domain W3C validator