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Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacval | Structured version Visualization version GIF version |
Description: The subfactorial is defined as the number of derangements (see derangval 34772) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.) |
Ref | Expression |
---|---|
derang.d | ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
subfac.n | ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) |
Ref | Expression |
---|---|
subfacval | ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7423 | . . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | fveq2d 6896 | . 2 ⊢ (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁))) |
3 | subfac.n | . 2 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) | |
4 | fvex 6905 | . 2 ⊢ (𝐷‘(1...𝑁)) ∈ V | |
5 | 2, 3, 4 | fvmpt 7000 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {cab 2705 ≠ wne 2936 ∀wral 3057 ↦ cmpt 5226 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7415 Fincfn 8958 1c1 11134 ℕ0cn0 12497 ...cfz 13511 ♯chash 14316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7418 |
This theorem is referenced by: derangen2 34779 subfaclefac 34781 subfac0 34782 subfac1 34783 subfacp1lem6 34790 |
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