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Theorem clwlkclwwlken 27542
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 6510 . . 3 (ClWalks‘𝐺) ∈ V
21rabex 5088 . 2 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∈ V
3 fvex 6510 . 2 (ClWWalks‘𝐺) ∈ V
4 2fveq3 6502 . . . . 5 (𝑤 = 𝑢 → (♯‘(1st𝑤)) = (♯‘(1st𝑢)))
54breq2d 4938 . . . 4 (𝑤 = 𝑢 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑢))))
65cbvrabv 3407 . . 3 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑢 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑢))}
7 fveq2 6497 . . . . 5 (𝑑 = 𝑐 → (2nd𝑑) = (2nd𝑐))
8 2fveq3 6502 . . . . . 6 (𝑑 = 𝑐 → (♯‘(2nd𝑑)) = (♯‘(2nd𝑐)))
98oveq1d 6990 . . . . 5 (𝑑 = 𝑐 → ((♯‘(2nd𝑑)) − 1) = ((♯‘(2nd𝑐)) − 1))
107, 9oveq12d 6993 . . . 4 (𝑑 = 𝑐 → ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
1110cbvmptv 5025 . . 3 (𝑑 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
126, 11clwlkclwwlkf1o 27541 . 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 8322 . 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 1445 1 (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2051  {crab 3087  Vcvv 3410   class class class wbr 4926  cmpt 5005  1-1-ontowf1o 6185  cfv 6186  (class class class)co 6975  1st c1st 7498  2nd c2nd 7499  cen 8302  1c1 10335  cle 10474  cmin 10669  chash 13504   prefix cpfx 13851  USPGraphcuspgr 26652  ClWalkscclwlks 27275  ClWWalkscclwwlk 27503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ifp 1045  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-int 4747  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-1o 7904  df-2o 7905  df-oadd 7908  df-er 8088  df-map 8207  df-pm 8208  df-en 8306  df-dom 8307  df-sdom 8308  df-fin 8309  df-dju 9123  df-card 9161  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-nn 11439  df-2 11502  df-n0 11707  df-xnn0 11779  df-z 11793  df-uz 12058  df-rp 12204  df-fz 12708  df-fzo 12849  df-hash 13505  df-word 13672  df-lsw 13725  df-concat 13733  df-s1 13758  df-substr 13803  df-pfx 13852  df-edg 26552  df-uhgr 26562  df-upgr 26586  df-uspgr 26654  df-wlks 27100  df-clwlks 27276  df-clwwlk 27504
This theorem is referenced by: (None)
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