Theorem derangval 33428

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 6756	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝑥))
2		f1oeq3 6757	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
3	1, 2	bitrd 278	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
4		raleq 3305	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦))
5	3, 4	anbi12d 631	. . . 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 2805	. . 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 6829	. 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 6838	. 2 ⊢ (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ∈ V
10	7, 8, 9	fvmpt 6931	1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2713   ≠ wne 2940  ∀wral 3061   ↦ cmpt 5175  –1-1-onto→wf1o 6478  ‘cfv 6479  Fincfn 8804  ♯chash 14145

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487

This theorem is referenced by:  derang0  33430  derangsn  33431  derangenlem  33432  subfaclefac  33437  subfacp1lem3  33443  subfacp1lem5  33445  subfacp1lem6  33446