Theorem derangval 35530

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 6799	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝑥))
2		f1oeq3 6800	. . . . . 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 3320	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦))
5	3, 4	anbi12d 643	. . . 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 2831	. . 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 6875	. 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 6884	. 2 ⊢ (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ∈ V
10	7, 8, 9	fvmpt 6979	1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  {cab 2743   ≠ wne 2960  ∀wral 3079   ↦ cmpt 5186  –1-1-onto→wf1o 6524  ‘cfv 6525  Fincfn 8931  ♯chash 14357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533

This theorem is referenced by:  derang0  35532  derangsn  35533  derangenlem  35534  subfaclefac  35539  subfacp1lem3  35545  subfacp1lem5  35547  subfacp1lem6  35548