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Theorem clwlkssizeeq 27880
 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.)
Assertion
Ref Expression
clwlkssizeeq ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}))
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem clwlkssizeeq
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 fvex 6659 . . . . 5 (ClWalks‘𝐺) ∈ V
21rabex 5200 . . . 4 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∈ V
32a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ∈ V)
4 eqid 2798 . . . 4 (1st𝑐) = (1st𝑐)
5 eqid 2798 . . . 4 (2nd𝑐) = (2nd𝑐)
6 eqid 2798 . . . 4 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
7 eqid 2798 . . . 4 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))
84, 5, 6, 7clwlknf1oclwwlkn 27879 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} ↦ ((2nd𝑐) prefix (♯‘(1st𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}–1-1-onto→(𝑁 ClWWalksN 𝐺))
93, 8hasheqf1od 13713 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}) = (♯‘(𝑁 ClWWalksN 𝐺)))
109eqcomd 2804 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441   ↦ cmpt 5111  ‘cfv 6325  (class class class)co 7136  1st c1st 7672  2nd c2nd 7673  ℕcn 11628  ♯chash 13689   prefix cpfx 14026  USPGraphcuspgr 26951  ClWalkscclwlks 27569   ClWWalksN cclwwlkn 27819 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-2o 8089  df-oadd 8092  df-er 8275  df-map 8394  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-dju 9317  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-n0 11889  df-xnn0 11959  df-z 11973  df-uz 12235  df-rp 12381  df-fz 12889  df-fzo 13032  df-hash 13690  df-word 13861  df-lsw 13909  df-concat 13917  df-s1 13944  df-substr 13997  df-pfx 14027  df-edg 26851  df-uhgr 26861  df-upgr 26885  df-uspgr 26953  df-wlks 27399  df-clwlks 27570  df-clwwlk 27777  df-clwwlkn 27820 This theorem is referenced by:  clwlksndivn  27881
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