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Theorem clwlkclwwlken 27795
 Description: The set of the nonempty closed walks and the set of closed walks as word are equinumerous in a simple pseudograph. (Contributed by AV, 25-May-2022.) (Proof shortened by AV, 4-Nov-2022.)
Assertion
Ref Expression
clwlkclwwlken (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))
Distinct variable group:   𝑤,𝐺

Proof of Theorem clwlkclwwlken
Dummy variables 𝑐 𝑑 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6665 . . 3 (ClWalks‘𝐺) ∈ V
21rabex 5211 . 2 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∈ V
3 fvex 6665 . 2 (ClWWalks‘𝐺) ∈ V
4 2fveq3 6657 . . . . 5 (𝑤 = 𝑢 → (♯‘(1st𝑤)) = (♯‘(1st𝑢)))
54breq2d 5054 . . . 4 (𝑤 = 𝑢 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑢))))
65cbvrabv 3467 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑢))}
7 fveq2 6652 . . . . 5 (𝑑 = 𝑐 → (2nd𝑑) = (2nd𝑐))
8 2fveq3 6657 . . . . . 6 (𝑑 = 𝑐 → (♯‘(2nd𝑑)) = (♯‘(2nd𝑐)))
98oveq1d 7155 . . . . 5 (𝑑 = 𝑐 → ((♯‘(2nd𝑑)) − 1) = ((♯‘(2nd𝑐)) − 1))
107, 9oveq12d 7158 . . . 4 (𝑑 = 𝑐 → ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
1110cbvmptv 5145 . . 3 (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
126, 11clwlkclwwlkf1o 27794 . 2 (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}–1-1-onto→(ClWWalks‘𝐺))
13 f1oen2g 8513 . 2 (({𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∈ V ∧ (ClWWalks‘𝐺) ∈ V ∧ (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}–1-1-onto→(ClWWalks‘𝐺)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))
142, 3, 12, 13mp3an12i 1462 1 (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2114  {crab 3134  Vcvv 3469   class class class wbr 5042   ↦ cmpt 5122  –1-1-onto→wf1o 6333  ‘cfv 6334  (class class class)co 7140  1st c1st 7673  2nd c2nd 7674   ≈ cen 8493  1c1 10527   ≤ cle 10665   − cmin 10859  ♯chash 13686   prefix cpfx 14023  USPGraphcuspgr 26939  ClWalkscclwlks 27557  ClWWalkscclwwlk 27764 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-lsw 13906  df-concat 13914  df-s1 13941  df-substr 13994  df-pfx 14024  df-edg 26839  df-uhgr 26849  df-upgr 26873  df-uspgr 26941  df-wlks 27387  df-clwlks 27558  df-clwwlk 27765 This theorem is referenced by: (None)
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