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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 32527) 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 7143 | . . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
2 | 1 | fveq2d 6649 | . 2 ⊢ (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁))) |
3 | subfac.n | . 2 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) | |
4 | fvex 6658 | . 2 ⊢ (𝐷‘(1...𝑁)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6745 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ≠ wne 2987 ∀wral 3106 ↦ cmpt 5110 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 1c1 10527 ℕ0cn0 11885 ...cfz 12885 ♯chash 13686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 |
This theorem is referenced by: derangen2 32534 subfaclefac 32536 subfac0 32537 subfac1 32538 subfacp1lem6 32545 |
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