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Mirrors > Home > MPE Home > Th. List > Mathboxes > derangen | Structured version Visualization version GIF version |
Description: The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.) |
Ref | Expression |
---|---|
derang.d | ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
Ref | Expression |
---|---|
derangen | ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) = (𝐷‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | derang.d | . . 3 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) | |
2 | 1 | derangenlem 32696 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ≤ (𝐷‘𝐵)) |
3 | ensym 8597 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
4 | 3 | adantr 484 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐵 ≈ 𝐴) |
5 | enfi 8778 | . . . 4 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin)) | |
6 | 5 | biimpar 481 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin) |
7 | 1 | derangenlem 32696 | . . 3 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ∈ Fin) → (𝐷‘𝐵) ≤ (𝐷‘𝐴)) |
8 | 4, 6, 7 | syl2anc 587 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐵) ≤ (𝐷‘𝐴)) |
9 | 1 | derangf 32693 | . . . . 5 ⊢ 𝐷:Fin⟶ℕ0 |
10 | 9 | ffvelrni 6854 | . . . 4 ⊢ (𝐴 ∈ Fin → (𝐷‘𝐴) ∈ ℕ0) |
11 | 6, 10 | syl 17 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) ∈ ℕ0) |
12 | 9 | ffvelrni 6854 | . . . 4 ⊢ (𝐵 ∈ Fin → (𝐷‘𝐵) ∈ ℕ0) |
13 | 12 | adantl 485 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐵) ∈ ℕ0) |
14 | nn0re 11978 | . . . 4 ⊢ ((𝐷‘𝐴) ∈ ℕ0 → (𝐷‘𝐴) ∈ ℝ) | |
15 | nn0re 11978 | . . . 4 ⊢ ((𝐷‘𝐵) ∈ ℕ0 → (𝐷‘𝐵) ∈ ℝ) | |
16 | letri3 10797 | . . . 4 ⊢ (((𝐷‘𝐴) ∈ ℝ ∧ (𝐷‘𝐵) ∈ ℝ) → ((𝐷‘𝐴) = (𝐷‘𝐵) ↔ ((𝐷‘𝐴) ≤ (𝐷‘𝐵) ∧ (𝐷‘𝐵) ≤ (𝐷‘𝐴)))) | |
17 | 14, 15, 16 | syl2an 599 | . . 3 ⊢ (((𝐷‘𝐴) ∈ ℕ0 ∧ (𝐷‘𝐵) ∈ ℕ0) → ((𝐷‘𝐴) = (𝐷‘𝐵) ↔ ((𝐷‘𝐴) ≤ (𝐷‘𝐵) ∧ (𝐷‘𝐵) ≤ (𝐷‘𝐴)))) |
18 | 11, 13, 17 | syl2anc 587 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → ((𝐷‘𝐴) = (𝐷‘𝐵) ↔ ((𝐷‘𝐴) ≤ (𝐷‘𝐵) ∧ (𝐷‘𝐵) ≤ (𝐷‘𝐴)))) |
19 | 2, 8, 18 | mpbir2and 713 | 1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ Fin) → (𝐷‘𝐴) = (𝐷‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {cab 2716 ≠ wne 2934 ∀wral 3053 class class class wbr 5027 ↦ cmpt 5107 –1-1-onto→wf1o 6332 ‘cfv 6333 ≈ cen 8545 Fincfn 8548 ℝcr 10607 ≤ cle 10747 ℕ0cn0 11969 ♯chash 13775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-oadd 8128 df-er 8313 df-map 8432 df-pm 8433 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-n0 11970 df-xnn0 12042 df-z 12056 df-uz 12318 df-fz 12975 df-hash 13776 |
This theorem is referenced by: derangen2 32699 |
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