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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 35172) 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 7439 | . . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁)) | |
| 2 | 1 | fveq2d 6910 | . 2 ⊢ (𝑛 = 𝑁 → (𝐷‘(1...𝑛)) = (𝐷‘(1...𝑁))) | 
| 3 | subfac.n | . 2 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) | |
| 4 | fvex 6919 | . 2 ⊢ (𝐷‘(1...𝑁)) ∈ V | |
| 5 | 2, 3, 4 | fvmpt 7016 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑆‘𝑁) = (𝐷‘(1...𝑁))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ↦ cmpt 5225 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 ℕ0cn0 12526 ...cfz 13547 ♯chash 14369 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: derangen2 35179 subfaclefac 35181 subfac0 35182 subfac1 35183 subfacp1lem6 35190 | 
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