Theorem derangval 31487

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 6270	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝑥))
2		f1oeq3 6271	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑓:𝐴–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
3	1, 2	bitrd 268	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑓:𝑥–1-1-onto→𝑥 ↔ 𝑓:𝐴–1-1-onto→𝐴))
4		raleq 3287	. . . . 5 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦 ↔ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦))
5	3, 4	anbi12d 616	. . . 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 2890	. . 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 6337	. 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 6344	. 2 ⊢ (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}) ∈ V
10	7, 8, 9	fvmpt 6426	1 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) = (♯‘{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑓‘𝑦) ≠ 𝑦)}))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {cab 2757   ≠ wne 2943  ∀wral 3061   ↦ cmpt 4864  –1-1-onto→wf1o 6029  ‘cfv 6030  Fincfn 8113  ♯chash 13321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038

This theorem is referenced by:  derang0  31489  derangsn  31490  derangenlem  31491  subfaclefac  31496  subfacp1lem3  31502  subfacp1lem5  31504  subfacp1lem6  31505