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Mirrors > Home > MPE Home > Th. List > Mathboxes > derangsn | Structured version Visualization version GIF version |
Description: The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.) |
Ref | Expression |
---|---|
derang.d | ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
Ref | Expression |
---|---|
derangsn | ⊢ (𝐴 ∈ 𝑉 → (𝐷‘{𝐴}) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snfi 8038 | . . . 4 ⊢ {𝐴} ∈ Fin | |
2 | derang.d | . . . . 5 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (#‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) | |
3 | 2 | derangval 31149 | . . . 4 ⊢ ({𝐴} ∈ Fin → (𝐷‘{𝐴}) = (#‘{𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)})) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (𝐷‘{𝐴}) = (#‘{𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)}) |
5 | f1of 6137 | . . . . . . . . . 10 ⊢ (𝑓:{𝐴}–1-1-onto→{𝐴} → 𝑓:{𝐴}⟶{𝐴}) | |
6 | 5 | adantr 481 | . . . . . . . . 9 ⊢ ((𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦) → 𝑓:{𝐴}⟶{𝐴}) |
7 | snidg 4206 | . . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
8 | ffvelrn 6357 | . . . . . . . . 9 ⊢ ((𝑓:{𝐴}⟶{𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓‘𝐴) ∈ {𝐴}) | |
9 | 6, 7, 8 | syl2anr 495 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)) → (𝑓‘𝐴) ∈ {𝐴}) |
10 | simpr 477 | . . . . . . . . . 10 ⊢ ((𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦) → ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦) | |
11 | fveq2 6191 | . . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑓‘𝑦) = (𝑓‘𝐴)) | |
12 | id 22 | . . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴) | |
13 | 11, 12 | neeq12d 2855 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → ((𝑓‘𝑦) ≠ 𝑦 ↔ (𝑓‘𝐴) ≠ 𝐴)) |
14 | 13 | rspcva 3307 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ {𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦) → (𝑓‘𝐴) ≠ 𝐴) |
15 | 7, 10, 14 | syl2an 494 | . . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)) → (𝑓‘𝐴) ≠ 𝐴) |
16 | nelsn 4212 | . . . . . . . . 9 ⊢ ((𝑓‘𝐴) ≠ 𝐴 → ¬ (𝑓‘𝐴) ∈ {𝐴}) | |
17 | 15, 16 | syl 17 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)) → ¬ (𝑓‘𝐴) ∈ {𝐴}) |
18 | 9, 17 | pm2.21dd 186 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)) → 𝑓 ∈ ∅) |
19 | 18 | ex 450 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ((𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦) → 𝑓 ∈ ∅)) |
20 | 19 | abssdv 3676 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)} ⊆ ∅) |
21 | ss0 3974 | . . . . 5 ⊢ ({𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)} ⊆ ∅ → {𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)} = ∅) | |
22 | 20, 21 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)} = ∅) |
23 | 22 | fveq2d 6195 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (#‘{𝑓 ∣ (𝑓:{𝐴}–1-1-onto→{𝐴} ∧ ∀𝑦 ∈ {𝐴} (𝑓‘𝑦) ≠ 𝑦)}) = (#‘∅)) |
24 | 4, 23 | syl5eq 2668 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐷‘{𝐴}) = (#‘∅)) |
25 | hash0 13158 | . 2 ⊢ (#‘∅) = 0 | |
26 | 24, 25 | syl6eq 2672 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐷‘{𝐴}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 {csn 4177 ↦ cmpt 4729 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 Fincfn 7955 0cc0 9936 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 |
This theorem is referenced by: subfac1 31160 |
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