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Mirrors > Home > MPE Home > Th. List > numclwwlk1lem2 | Structured version Visualization version GIF version |
Description: The set of double loops of length 𝑁 on vertex 𝑋 and the set of closed walks of length less by 2 on 𝑋 combined with the neighbors of 𝑋 are equinumerous. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Jul-2022.) (Proof shortened by AV, 3-Nov-2022.) |
Ref | Expression |
---|---|
extwwlkfab.v | ⊢ 𝑉 = (Vtx‘𝐺) |
extwwlkfab.c | ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) |
extwwlkfab.f | ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) |
Ref | Expression |
---|---|
numclwwlk1lem2 | ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | extwwlkfab.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | extwwlkfab.c | . . 3 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) | |
3 | extwwlkfab.f | . . 3 ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) | |
4 | oveq1 7209 | . . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥 prefix (𝑁 − 2)) = (𝑢 prefix (𝑁 − 2))) | |
5 | fveq1 6705 | . . . . 5 ⊢ (𝑥 = 𝑢 → (𝑥‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))) | |
6 | 4, 5 | opeq12d 4782 | . . . 4 ⊢ (𝑥 = 𝑢 → 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉 = 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉) |
7 | 6 | cbvmptv 5147 | . . 3 ⊢ (𝑥 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉) = (𝑢 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))〉) |
8 | 1, 2, 3, 7 | numclwwlk1lem2f1o 28414 | . 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑥 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉):(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋))) |
9 | ovex 7235 | . . 3 ⊢ (𝑋𝐶𝑁) ∈ V | |
10 | 9 | f1oen 8638 | . 2 ⊢ ((𝑥 ∈ (𝑋𝐶𝑁) ↦ 〈(𝑥 prefix (𝑁 − 2)), (𝑥‘(𝑁 − 1))〉):(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) |
11 | 8, 10 | syl 17 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 {crab 3058 〈cop 4537 class class class wbr 5043 ↦ cmpt 5124 × cxp 5538 –1-1-onto→wf1o 6368 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 ≈ cen 8612 1c1 10713 − cmin 11045 2c2 11868 3c3 11869 ℤ≥cuz 12421 prefix cpfx 14218 Vtxcvtx 27059 USGraphcusgr 27212 NeighbVtx cnbgr 27392 ClWWalksNOncclwwlknon 28142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-hash 13880 df-word 14053 df-lsw 14101 df-concat 14109 df-s1 14136 df-substr 14189 df-pfx 14219 df-s2 14396 df-edg 27111 df-upgr 27145 df-umgr 27146 df-usgr 27214 df-nbgr 27393 df-wwlks 27886 df-wwlksn 27887 df-clwwlk 28037 df-clwwlkn 28080 df-clwwlknon 28143 |
This theorem is referenced by: numclwwlk1 28416 |
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