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Mirrors > Home > MPE Home > Th. List > clwwlknonfin | Structured version Visualization version GIF version |
Description: In a finite graph 𝐺, the set of closed walks on vertex 𝑋 of length 𝑁 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknonfin.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
clwwlknonfin | ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlknon 27634 | . 2 ⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} | |
2 | clwwlknonfin.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | eleq1i 2851 | . . . 4 ⊢ (𝑉 ∈ Fin ↔ (Vtx‘𝐺) ∈ Fin) |
4 | clwwlknfi 27577 | . . . 4 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) | |
5 | 3, 4 | sylbi 209 | . . 3 ⊢ (𝑉 ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
6 | rabfi 8537 | . . 3 ⊢ ((𝑁 ClWWalksN 𝐺) ∈ Fin → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑉 ∈ Fin → {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∈ Fin) |
8 | 1, 7 | syl5eqel 2865 | 1 ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 {crab 3087 ‘cfv 6186 (class class class)co 6975 Fincfn 8305 0cc0 10334 Vtxcvtx 26500 ClWWalksN cclwwlkn 27555 ClWWalksNOncclwwlknon 27631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-dju 9123 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-n0 11707 df-xnn0 11779 df-z 11793 df-uz 12058 df-fz 12708 df-fzo 12849 df-seq 13184 df-exp 13244 df-hash 13505 df-word 13672 df-clwwlk 27504 df-clwwlkn 27556 df-clwwlknon 27632 |
This theorem is referenced by: numclwwlk1 27925 clwlknon2num 27937 numclwlk1lem2 27939 numclwwlk3lem2 27957 numclwwlk3 27958 numclwwlk4 27959 numclwwlk5 27961 numclwwlk6 27963 |
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