Theorem derangval 35361

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 6763	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝑥))
2		f1oeq3 6764	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
3	1, 2	bitrd 279	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
4		raleq 3293	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦))
5	3, 4	anbi12d 632	. . . 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 2802	. . 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 6838	. 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 6847	. 2 ⊢ (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ∈ V
10	7, 8, 9	fvmpt 6941	1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051   ↦ cmpt 5179  –1-1-onto→wf1o 6491  ‘cfv 6492  Fincfn 8883  ♯chash 14253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  derang0  35363  derangsn  35364  derangenlem  35365  subfaclefac  35370  subfacp1lem3  35376  subfacp1lem5  35378  subfacp1lem6  35379