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Theorem derangval 34995
Description: Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)
Hypothesis
Ref Expression
derang.d 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
Assertion
Ref Expression
derangval (𝐴 ∈ Fin → (𝐷𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}))
Distinct variable group:   𝑥,𝑓,𝑦,𝐴
Allowed substitution hints:   𝐷(𝑥,𝑦,𝑓)

Proof of Theorem derangval
StepHypRef Expression
1 f1oeq2 6832 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑥𝑓:𝐴1-1-onto𝑥))
2 f1oeq3 6833 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1-onto𝐴))
31, 2bitrd 278 . . . . 5 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑥𝑓:𝐴1-1-onto𝐴))
4 raleq 3312 . . . . 5 (𝑥 = 𝐴 → (∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦 ↔ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦))
53, 4anbi12d 630 . . . 4 (𝑥 = 𝐴 → ((𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦) ↔ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)))
65abbidv 2795 . . 3 (𝑥 = 𝐴 → {𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)} = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)})
76fveq2d 6905 . 2 (𝑥 = 𝐴 → (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}) = (♯‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}))
8 derang.d . 2 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)}))
9 fvex 6914 . 2 (♯‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}) ∈ V
107, 8, 9fvmpt 7009 1 (𝐴 ∈ Fin → (𝐷𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {cab 2703  wne 2930  wral 3051  cmpt 5236  1-1-ontowf1o 6553  cfv 6554  Fincfn 8974  chash 14347
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 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562
This theorem is referenced by:  derang0  34997  derangsn  34998  derangenlem  34999  subfaclefac  35004  subfacp1lem3  35010  subfacp1lem5  35012  subfacp1lem6  35013
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