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Mirrors > Home > MPE Home > Th. List > clwlkssizeeq | Structured version Visualization version GIF version |
Description: The size of the set of closed walks as words of length 𝑁 corresponds to the size of the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) (Revised by AV, 26-May-2022.) (Proof shortened by AV, 3-Nov-2022.) |
Ref | Expression |
---|---|
clwlkssizeeq | ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6852 | . . . . 5 ⊢ (ClWalks‘𝐺) ∈ V | |
2 | 1 | rabex 5287 | . . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ∈ V) |
4 | eqid 2737 | . . . 4 ⊢ (1st ‘𝑐) = (1st ‘𝑐) | |
5 | eqid 2737 | . . . 4 ⊢ (2nd ‘𝑐) = (2nd ‘𝑐) | |
6 | eqid 2737 | . . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} | |
7 | eqid 2737 | . . . 4 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) | |
8 | 4, 5, 6, 7 | clwlknf1oclwwlkn 28857 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺)) |
9 | 3, 8 | hasheqf1od 14207 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}) = (♯‘(𝑁 ClWWalksN 𝐺))) |
10 | 9 | eqcomd 2743 | 1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3405 Vcvv 3443 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 1st c1st 7911 2nd c2nd 7912 ℕcn 12111 ♯chash 14184 prefix cpfx 14516 USPGraphcuspgr 27928 ClWalkscclwlks 28547 ClWWalksN cclwwlkn 28797 |
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 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 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 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-oadd 8408 df-er 8606 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-xnn0 12444 df-z 12458 df-uz 12722 df-rp 12870 df-fz 13379 df-fzo 13522 df-hash 14185 df-word 14357 df-lsw 14405 df-concat 14413 df-s1 14438 df-substr 14487 df-pfx 14517 df-edg 27828 df-uhgr 27838 df-upgr 27862 df-uspgr 27930 df-wlks 28376 df-clwlks 28548 df-clwwlk 28755 df-clwwlkn 28798 |
This theorem is referenced by: clwlksndivn 28859 |
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