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Mirrors > Home > MPE Home > Th. List > clwwlknon | Structured version Visualization version GIF version |
Description: The set of closed walks on vertex 𝑋 of length 𝑁 in a graph 𝐺 as words over the set of vertices. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknon | ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2749 | . . . 4 ⊢ (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋)) | |
2 | 1 | rabbidv 3413 | . . 3 ⊢ (𝑣 = 𝑋 → {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
3 | oveq1 7353 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 ClWWalksN 𝐺) = (𝑁 ClWWalksN 𝐺)) | |
4 | 3 | rabeqdv 3420 | . . 3 ⊢ (𝑛 = 𝑁 → {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
5 | clwwlknonmpo 28807 | . . 3 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) | |
6 | ovex 7379 | . . . 4 ⊢ (𝑁 ClWWalksN 𝐺) ∈ V | |
7 | 6 | rabex 5284 | . . 3 ⊢ {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ V |
8 | 2, 4, 5, 7 | ovmpo 7504 | . 2 ⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
9 | 5 | mpondm0 7581 | . . 3 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅) |
10 | isclwwlkn 28745 | . . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑤 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑤) = 𝑁)) | |
11 | eqid 2737 | . . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
12 | 11 | clwwlkbp 28703 | . . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅)) |
13 | fstwrdne 14367 | . . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅) → (𝑤‘0) ∈ (Vtx‘𝐺)) | |
14 | 13 | 3adant1 1130 | . . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ V ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 𝑤 ≠ ∅) → (𝑤‘0) ∈ (Vtx‘𝐺)) |
15 | 12, 14 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝑤 ∈ (ClWWalks‘𝐺) → (𝑤‘0) ∈ (Vtx‘𝐺)) |
16 | 15 | adantr 482 | . . . . . . . . . . 11 ⊢ ((𝑤 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑤) = 𝑁) → (𝑤‘0) ∈ (Vtx‘𝐺)) |
17 | 10, 16 | sylbi 216 | . . . . . . . . . 10 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → (𝑤‘0) ∈ (Vtx‘𝐺)) |
18 | 17 | adantr 482 | . . . . . . . . 9 ⊢ ((𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) ∈ (Vtx‘𝐺)) |
19 | eleq1 2825 | . . . . . . . . . 10 ⊢ ((𝑤‘0) = 𝑋 → ((𝑤‘0) ∈ (Vtx‘𝐺) ↔ 𝑋 ∈ (Vtx‘𝐺))) | |
20 | 19 | adantl 483 | . . . . . . . . 9 ⊢ ((𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → ((𝑤‘0) ∈ (Vtx‘𝐺) ↔ 𝑋 ∈ (Vtx‘𝐺))) |
21 | 18, 20 | mpbid 231 | . . . . . . . 8 ⊢ ((𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → 𝑋 ∈ (Vtx‘𝐺)) |
22 | clwwlknnn 28751 | . . . . . . . . . 10 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 ∈ ℕ) | |
23 | 22 | nnnn0d 12403 | . . . . . . . . 9 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 ∈ ℕ0) |
24 | 23 | adantr 482 | . . . . . . . 8 ⊢ ((𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → 𝑁 ∈ ℕ0) |
25 | 21, 24 | jca 513 | . . . . . . 7 ⊢ ((𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) → (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0)) |
26 | 25 | ex 414 | . . . . . 6 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑤‘0) = 𝑋 → (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0))) |
27 | 26 | con3rr3 155 | . . . . 5 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → ¬ (𝑤‘0) = 𝑋)) |
28 | 27 | ralrimiv 3140 | . . . 4 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ (𝑤‘0) = 𝑋) |
29 | rabeq0 4339 | . . . 4 ⊢ ({𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = ∅ ↔ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺) ¬ (𝑤‘0) = 𝑋) | |
30 | 28, 29 | sylibr 233 | . . 3 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = ∅) |
31 | 9, 30 | eqtr4d 2780 | . 2 ⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}) |
32 | 8, 31 | pm2.61i 182 | 1 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∀wral 3062 {crab 3405 Vcvv 3443 ∅c0 4277 ‘cfv 6488 (class class class)co 7346 0cc0 10981 ℕ0cn0 12343 ♯chash 14154 Word cword 14326 Vtxcvtx 27721 ClWWalkscclwwlk 28699 ClWWalksN cclwwlkn 28742 ClWWalksNOncclwwlknon 28805 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5237 ax-sep 5251 ax-nul 5258 ax-pow 5315 ax-pr 5379 ax-un 7659 ax-cnex 11037 ax-resscn 11038 ax-1cn 11039 ax-icn 11040 ax-addcl 11041 ax-addrcl 11042 ax-mulcl 11043 ax-mulrcl 11044 ax-mulcom 11045 ax-addass 11046 ax-mulass 11047 ax-distr 11048 ax-i2m1 11049 ax-1ne0 11050 ax-1rid 11051 ax-rnegex 11052 ax-rrecex 11053 ax-cnre 11054 ax-pre-lttri 11055 ax-pre-lttrn 11056 ax-pre-ltadd 11057 ax-pre-mulgt0 11058 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3735 df-csb 3851 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3924 df-nul 4278 df-if 4482 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4861 df-int 4903 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5184 df-tr 5218 df-id 5525 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5582 df-we 5584 df-xp 5633 df-rel 5634 df-cnv 5635 df-co 5636 df-dm 5637 df-rn 5638 df-res 5639 df-ima 5640 df-pred 6246 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6440 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7302 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7790 df-1st 7908 df-2nd 7909 df-frecs 8176 df-wrecs 8207 df-recs 8281 df-rdg 8320 df-1o 8376 df-oadd 8380 df-er 8578 df-map 8697 df-en 8814 df-dom 8815 df-sdom 8816 df-fin 8817 df-card 9805 df-pnf 11121 df-mnf 11122 df-xr 11123 df-ltxr 11124 df-le 11125 df-sub 11317 df-neg 11318 df-nn 12084 df-n0 12344 df-xnn0 12416 df-z 12430 df-uz 12693 df-fz 13350 df-fzo 13493 df-hash 14155 df-word 14327 df-clwwlk 28700 df-clwwlkn 28743 df-clwwlknon 28806 |
This theorem is referenced by: isclwwlknon 28809 clwwlknonfin 28812 clwwlknon1 28815 clwwlknon2 28820 clwwlknondisj 28829 clwwlkvbij 28831 extwwlkfab 29070 clwwlknonclwlknonf1o 29080 |
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