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Theorem clwlkssizeeqOLD 27257
 Description: Obsolete version of clwlkssizeeq 27256 as of 26-May-2022. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
clwlkssizeeqOLD ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐

Proof of Theorem clwlkssizeeqOLD
Dummy variables 𝑤 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6344 . . . . 5 (ClWalks‘𝐺) ∈ V
21rabex 4947 . . . 4 {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} ∈ V
32a1i 11 . . 3 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} ∈ V)
4 eqid 2771 . . . 4 (1st𝑤) = (1st𝑤)
5 eqid 2771 . . . 4 (2nd𝑤) = (2nd𝑤)
6 fveq2 6333 . . . . . . 7 (𝑐 = 𝑤 → (1st𝑐) = (1st𝑤))
76fveq2d 6337 . . . . . 6 (𝑐 = 𝑤 → (♯‘(1st𝑐)) = (♯‘(1st𝑤)))
87eqeq1d 2773 . . . . 5 (𝑐 = 𝑤 → ((♯‘(1st𝑐)) = 𝑁 ↔ (♯‘(1st𝑤)) = 𝑁))
98cbvrabv 3349 . . . 4 {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
10 fveq2 6333 . . . . . 6 (𝑢 = 𝑤 → (2nd𝑢) = (2nd𝑤))
11 fveq2 6333 . . . . . . . 8 (𝑢 = 𝑤 → (1st𝑢) = (1st𝑤))
1211fveq2d 6337 . . . . . . 7 (𝑢 = 𝑤 → (♯‘(1st𝑢)) = (♯‘(1st𝑤)))
1312opeq2d 4547 . . . . . 6 (𝑢 = 𝑤 → ⟨0, (♯‘(1st𝑢))⟩ = ⟨0, (♯‘(1st𝑤))⟩)
1410, 13oveq12d 6814 . . . . 5 (𝑢 = 𝑤 → ((2nd𝑢) substr ⟨0, (♯‘(1st𝑢))⟩) = ((2nd𝑤) substr ⟨0, (♯‘(1st𝑤))⟩))
1514cbvmptv 4885 . . . 4 (𝑢 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} ↦ ((2nd𝑢) substr ⟨0, (♯‘(1st𝑢))⟩)) = (𝑤 ∈ {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} ↦ ((2nd𝑤) substr ⟨0, (♯‘(1st𝑤))⟩))
164, 5, 9, 15clwlksf1oclwwlkOLD 27251 . . 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 13346 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}) = (♯‘(𝑁 ClWWalksN 𝐺)))
1817eqcomd 2777 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065  Vcvv 3351  ⟨cop 4323   ↦ cmpt 4864  ‘cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  0cc0 10142  ♯chash 13321   substr csubstr 13491  ℙcprime 15592  FinUSGraphcfusgr 26431  ClWalkscclwlks 26901   ClWWalksN cclwwlkn 27174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-prm 15593  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-usgr 26268  df-fusgr 26432  df-wlks 26730  df-clwlks 26902  df-clwwlk 27132  df-clwwlkn 27176 This theorem is referenced by: (None)
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