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Theorem clwlkclwwlken 28355
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 6781 . . 3 (ClWalks‘𝐺) ∈ V
21rabex 5259 . 2 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∈ V
3 fvex 6781 . 2 (ClWWalks‘𝐺) ∈ V
4 2fveq3 6773 . . . . 5 (𝑤 = 𝑢 → (♯‘(1st𝑤)) = (♯‘(1st𝑢)))
54breq2d 5090 . . . 4 (𝑤 = 𝑢 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑢))))
65cbvrabv 3424 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑢))}
7 fveq2 6768 . . . . 5 (𝑑 = 𝑐 → (2nd𝑑) = (2nd𝑐))
8 2fveq3 6773 . . . . . 6 (𝑑 = 𝑐 → (♯‘(2nd𝑑)) = (♯‘(2nd𝑐)))
98oveq1d 7283 . . . . 5 (𝑑 = 𝑐 → ((♯‘(2nd𝑑)) − 1) = ((♯‘(2nd𝑐)) − 1))
107, 9oveq12d 7286 . . . 4 (𝑑 = 𝑐 → ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
1110cbvmptv 5191 . . 3 (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
126, 11clwlkclwwlkf1o 28354 . 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 8727 . 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 1463 1 (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2109  {crab 3069  Vcvv 3430   class class class wbr 5078  cmpt 5161  1-1-ontowf1o 6429  cfv 6430  (class class class)co 7268  1st c1st 7815  2nd c2nd 7816  cen 8704  1c1 10856  cle 10994  cmin 11188  chash 14025   prefix cpfx 14364  USPGraphcuspgr 27499  ClWalkscclwlks 28117  ClWWalkscclwwlk 28324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-cnex 10911  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931  ax-pre-mulgt0 10932
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-int 4885  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-1o 8281  df-2o 8282  df-oadd 8285  df-er 8472  df-map 8591  df-pm 8592  df-en 8708  df-dom 8709  df-sdom 8710  df-fin 8711  df-dju 9643  df-card 9681  df-pnf 10995  df-mnf 10996  df-xr 10997  df-ltxr 10998  df-le 10999  df-sub 11190  df-neg 11191  df-nn 11957  df-2 12019  df-n0 12217  df-xnn0 12289  df-z 12303  df-uz 12565  df-rp 12713  df-fz 13222  df-fzo 13365  df-hash 14026  df-word 14199  df-lsw 14247  df-concat 14255  df-s1 14282  df-substr 14335  df-pfx 14365  df-edg 27399  df-uhgr 27409  df-upgr 27433  df-uspgr 27501  df-wlks 27947  df-clwlks 28118  df-clwwlk 28325
This theorem is referenced by: (None)
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