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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 6682 | . . . . 5 ⊢ (ClWalks‘𝐺) ∈ V | |
2 | 1 | rabex 5234 | . . . 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 2821 | . . . 4 ⊢ (1st ‘𝑐) = (1st ‘𝑐) | |
5 | eqid 2821 | . . . 4 ⊢ (2nd ‘𝑐) = (2nd ‘𝑐) | |
6 | eqid 2821 | . . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} | |
7 | eqid 2821 | . . . 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 27862 | . . 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 13713 | . 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}) = (♯‘(𝑁 ClWWalksN 𝐺))) |
10 | 9 | eqcomd 2827 | 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 398 = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3494 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 ℕcn 11637 ♯chash 13689 prefix cpfx 14031 USPGraphcuspgr 26932 ClWalkscclwlks 27550 ClWWalksN cclwwlkn 27801 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-xnn0 11967 df-z 11981 df-uz 12243 df-rp 12389 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-lsw 13914 df-concat 13922 df-s1 13949 df-substr 14002 df-pfx 14032 df-edg 26832 df-uhgr 26842 df-upgr 26866 df-uspgr 26934 df-wlks 27380 df-clwlks 27551 df-clwwlk 27759 df-clwwlkn 27802 |
This theorem is referenced by: clwlksndivn 27864 |
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