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Mirrors > Home > MPE Home > Th. List > clwwlknfi | Structured version Visualization version GIF version |
Description: If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) (Proof shortened by JJ, 18-Nov-2022.) |
Ref | Expression |
---|---|
clwwlknfi | ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlkn 30054 | . 2 ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} | |
2 | wrdnfi 14582 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) | |
3 | clwwlksswrd 30015 | . . . 4 ⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | |
4 | rabss2 4087 | . . . 4 ⊢ ((ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) | |
5 | 3, 4 | mp1i 13 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
6 | 2, 5 | ssfid 9298 | . 2 ⊢ ((Vtx‘𝐺) ∈ Fin → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) |
7 | 1, 6 | eqeltrid 2842 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 {crab 3432 ⊆ wss 3962 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 ♯chash 14365 Word cword 14548 Vtxcvtx 29027 ClWWalkscclwwlk 30009 ClWWalksN cclwwlkn 30052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-word 14549 df-clwwlk 30010 df-clwwlkn 30053 |
This theorem is referenced by: qerclwwlknfi 30101 hashclwwlkn0 30102 clwwlknonfin 30122 |
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