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Theorem clwlkssizeeq 26837
Description: The size of the set of closed walks (defined as words) of length n corresponds to the size of the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.)
Assertion
Ref Expression
clwlkssizeeq ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘(𝑁 ClWWalksN 𝐺)) = (#‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐

Proof of Theorem clwlkssizeeq
Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6158 . . . . 5 (ClWalks‘𝐺) ∈ V
21rabex 4773 . . . 4 {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V
32a1i 11 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V)
4 eqid 2621 . . . 4 (1st𝑤) = (1st𝑤)
5 eqid 2621 . . . 4 (2nd𝑤) = (2nd𝑤)
6 fveq2 6148 . . . . . . 7 (𝑐 = 𝑤 → (1st𝑐) = (1st𝑤))
76fveq2d 6152 . . . . . 6 (𝑐 = 𝑤 → (#‘(1st𝑐)) = (#‘(1st𝑤)))
87eqeq1d 2623 . . . . 5 (𝑐 = 𝑤 → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st𝑤)) = 𝑁))
98cbvrabv 3185 . . . 4 {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑤)) = 𝑁}
10 fveq2 6148 . . . . . 6 (𝑢 = 𝑤 → (2nd𝑢) = (2nd𝑤))
11 fveq2 6148 . . . . . . . 8 (𝑢 = 𝑤 → (1st𝑢) = (1st𝑤))
1211fveq2d 6152 . . . . . . 7 (𝑢 = 𝑤 → (#‘(1st𝑢)) = (#‘(1st𝑤)))
1312opeq2d 4377 . . . . . 6 (𝑢 = 𝑤 → ⟨0, (#‘(1st𝑢))⟩ = ⟨0, (#‘(1st𝑤))⟩)
1410, 13oveq12d 6622 . . . . 5 (𝑢 = 𝑤 → ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩))
1514cbvmptv 4710 . . . 4 (𝑢 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ↦ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩)) = (𝑤 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ↦ ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩))
164, 5, 9, 15clwlksf1oclwwlk 26836 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (𝑢 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ↦ ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩)):{𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
173, 16hasheqf1od 13084 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}) = (#‘(𝑁 ClWWalksN 𝐺)))
1817eqcomd 2627 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘(𝑁 ClWWalksN 𝐺)) = (#‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cop 4154  cmpt 4673  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880  #chash 13057   substr csubstr 13234  cprime 15309   FinUSGraph cfusgr 26096  ClWalkscclwlks 26535   ClWWalksN cclwwlksn 26743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-xnn0 11308  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-dvds 14908  df-prm 15310  df-edg 25840  df-uhgr 25849  df-upgr 25873  df-uspgr 25938  df-usgr 25939  df-fusgr 26097  df-wlks 26365  df-clwlks 26536  df-clwwlks 26744  df-clwwlksn 26745
This theorem is referenced by:  clwlksndivn  26838
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