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Theorem clwwlknonmpo 28862
Description: (ClWWalksNOnβ€˜πΊ) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)
Assertion
Ref Expression
clwwlknonmpo (ClWWalksNOnβ€˜πΊ) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣})
Distinct variable group:   𝑛,𝐺,𝑣,𝑀

Proof of Theorem clwwlknonmpo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6839 . . . 4 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
2 eqidd 2738 . . . 4 (𝑔 = 𝐺 β†’ β„•0 = β„•0)
3 oveq2 7359 . . . . 5 (𝑔 = 𝐺 β†’ (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
43rabeqdv 3420 . . . 4 (𝑔 = 𝐺 β†’ {𝑀 ∈ (𝑛 ClWWalksN 𝑔) ∣ (π‘€β€˜0) = 𝑣} = {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣})
51, 2, 4mpoeq123dv 7426 . . 3 (𝑔 = 𝐺 β†’ (𝑣 ∈ (Vtxβ€˜π‘”), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝑔) ∣ (π‘€β€˜0) = 𝑣}) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}))
6 df-clwwlknon 28861 . . 3 ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtxβ€˜π‘”), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝑔) ∣ (π‘€β€˜0) = 𝑣}))
7 fvex 6852 . . . 4 (Vtxβ€˜πΊ) ∈ V
8 nn0ex 12377 . . . 4 β„•0 ∈ V
97, 8mpoex 8004 . . 3 (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}) ∈ V
105, 6, 9fvmpt 6945 . 2 (𝐺 ∈ V β†’ (ClWWalksNOnβ€˜πΊ) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}))
11 fvprc 6831 . . 3 (Β¬ 𝐺 ∈ V β†’ (ClWWalksNOnβ€˜πΊ) = βˆ…)
12 fvprc 6831 . . . . 5 (Β¬ 𝐺 ∈ V β†’ (Vtxβ€˜πΊ) = βˆ…)
1312orcd 871 . . . 4 (Β¬ 𝐺 ∈ V β†’ ((Vtxβ€˜πΊ) = βˆ… ∨ β„•0 = βˆ…))
14 0mpo0 7434 . . . 4 (((Vtxβ€˜πΊ) = βˆ… ∨ β„•0 = βˆ…) β†’ (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}) = βˆ…)
1513, 14syl 17 . . 3 (Β¬ 𝐺 ∈ V β†’ (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}) = βˆ…)
1611, 15eqtr4d 2780 . 2 (Β¬ 𝐺 ∈ V β†’ (ClWWalksNOnβ€˜πΊ) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}))
1710, 16pm2.61i 182 1 (ClWWalksNOnβ€˜πΊ) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣})
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {crab 3405  Vcvv 3443  βˆ…c0 4280  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  0cc0 11009  β„•0cn0 12371  Vtxcvtx 27776   ClWWalksN cclwwlkn 28797  ClWWalksNOncclwwlknon 28860
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-1cn 11067  ax-addcl 11069
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-nn 12112  df-n0 12372  df-clwwlknon 28861
This theorem is referenced by:  clwwlknon  28863  clwwlk0on0  28865  clwwlknon0  28866  2clwwlk2clwwlklem  29119
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