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Theorem clwlkssizeeq 41278
 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 6113 . . . . 5 (ClWalkS‘𝐺) ∈ V
21rabex 4740 . . . 4 {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V
32a1i 11 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} ∈ V)
4 eqid 2610 . . . 4 (1st𝑤) = (1st𝑤)
5 eqid 2610 . . . 4 (2nd𝑤) = (2nd𝑤)
6 fveq2 6103 . . . . . . 7 (𝑐 = 𝑤 → (1st𝑐) = (1st𝑤))
76fveq2d 6107 . . . . . 6 (𝑐 = 𝑤 → (#‘(1st𝑐)) = (#‘(1st𝑤)))
87eqeq1d 2612 . . . . 5 (𝑐 = 𝑤 → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st𝑤)) = 𝑁))
98cbvrabv 3172 . . . 4 {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁} = {𝑤 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st𝑤)) = 𝑁}
10 fveq2 6103 . . . . . 6 (𝑢 = 𝑤 → (2nd𝑢) = (2nd𝑤))
11 fveq2 6103 . . . . . . . 8 (𝑢 = 𝑤 → (1st𝑢) = (1st𝑤))
1211fveq2d 6107 . . . . . . 7 (𝑢 = 𝑤 → (#‘(1st𝑢)) = (#‘(1st𝑤)))
1312opeq2d 4347 . . . . . 6 (𝑢 = 𝑤 → ⟨0, (#‘(1st𝑢))⟩ = ⟨0, (#‘(1st𝑤))⟩)
1410, 13oveq12d 6567 . . . . 5 (𝑢 = 𝑤 → ((2nd𝑢) substr ⟨0, (#‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (#‘(1st𝑤))⟩))
1514cbvmptv 4678 . . . 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 41277 . . 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 13006 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘{𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}) = (#‘(𝑁 ClWWalkSN 𝐺)))
1817eqcomd 2616 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘(𝑁 ClWWalkSN 𝐺)) = (#‘{𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  #chash 12979   substr csubstr 13150  ℙcprime 15223   FinUSGraph cfusgr 40535  ClWalkScclwlks 40976   ClWWalkSN cclwwlksn 41184 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-1wlks 40800  df-wlks 40801  df-clwlks 40977  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  clwlksndivn  41279
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