Theorem derangval 35680

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:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))

Assertion

Ref	Expression
derangval	⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))

Distinct variable group:   𝑥,𝑓,𝑦,𝐴

Allowed substitution hints:   𝐷(𝑥, 𝑦, 𝑓)

Proof of Theorem derangval

Step	Hyp	Ref	Expression
1		f1oeq2 6816	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝑥))
2		f1oeq3 6817	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
3	1, 2	bitrd 282	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
4		raleq 3323	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦))
5	3, 4	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦) ↔ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)))
6	5	abbidv 2832	. . 3 ⊢ (𝑥 = 𝐴 → {𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)} = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)})
7	6	fveq2d 6892	. 2 ⊢ (𝑥 = 𝐴 → (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))
8		derang.d	. 2 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)}))
9		fvex 6901	. 2 ⊢ (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ∈ V
10	7, 8, 9	fvmpt 6996	1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744   ≠ wne 2961  ∀wral 3082   ↦ cmpt 5197  –1-1-onto→wf1o 6542  ‘cfv 6543  Fincfn 8952  ♯chash 14386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551

This theorem is used by:  derang0  35682  derangsn  35683  derangenlem  35684  subfaclefac  35689  subfacp1lem3  35695  subfacp1lem5  35697  subfacp1lem6  35698