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Theorem derangval 30403
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 6041 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑥𝑓:𝐴1-1-onto𝑥))
2 f1oeq3 6042 . . . . . 6 (𝑥 = 𝐴 → (𝑓:𝐴1-1-onto𝑥𝑓:𝐴1-1-onto𝐴))
31, 2bitrd 267 . . . . 5 (𝑥 = 𝐴 → (𝑓:𝑥1-1-onto𝑥𝑓:𝐴1-1-onto𝐴))
4 raleq 3115 . . . . 5 (𝑥 = 𝐴 → (∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦 ↔ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦))
53, 4anbi12d 743 . . . 4 (𝑥 = 𝐴 → ((𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦) ↔ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)))
65abbidv 2728 . . 3 (𝑥 = 𝐴 → {𝑓 ∣ (𝑓:𝑥1-1-onto𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) ≠ 𝑦)} = {𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)})
76fveq2d 6107 . 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 6113 . 2 (#‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}) ∈ V
107, 8, 9fvmpt 6191 1 (𝐴 ∈ Fin → (𝐷𝐴) = (#‘{𝑓 ∣ (𝑓:𝐴1-1-onto𝐴 ∧ ∀𝑦𝐴 (𝑓𝑦) ≠ 𝑦)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  {cab 2596  wne 2780  wral 2896  cmpt 4643  1-1-ontowf1o 5803  cfv 5804  Fincfn 7841  #chash 12979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812
This theorem is referenced by:  derang0  30405  derangsn  30406  derangenlem  30407  subfaclefac  30412  subfacp1lem3  30418  subfacp1lem5  30420  subfacp1lem6  30421
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