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Theorem clwwlknonmpo 29886
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 6891 . . . 4 (𝑔 = 𝐺 β†’ (Vtxβ€˜π‘”) = (Vtxβ€˜πΊ))
2 eqidd 2728 . . . 4 (𝑔 = 𝐺 β†’ β„•0 = β„•0)
3 oveq2 7422 . . . . 5 (𝑔 = 𝐺 β†’ (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
43rabeqdv 3442 . . . 4 (𝑔 = 𝐺 β†’ {𝑀 ∈ (𝑛 ClWWalksN 𝑔) ∣ (π‘€β€˜0) = 𝑣} = {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣})
51, 2, 4mpoeq123dv 7489 . . 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 29885 . . 3 ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtxβ€˜π‘”), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝑔) ∣ (π‘€β€˜0) = 𝑣}))
7 fvex 6904 . . . 4 (Vtxβ€˜πΊ) ∈ V
8 nn0ex 12500 . . . 4 β„•0 ∈ V
97, 8mpoex 8078 . . 3 (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}) ∈ V
105, 6, 9fvmpt 6999 . 2 (𝐺 ∈ V β†’ (ClWWalksNOnβ€˜πΊ) = (𝑣 ∈ (Vtxβ€˜πΊ), 𝑛 ∈ β„•0 ↦ {𝑀 ∈ (𝑛 ClWWalksN 𝐺) ∣ (π‘€β€˜0) = 𝑣}))
11 fvprc 6883 . . 3 (Β¬ 𝐺 ∈ V β†’ (ClWWalksNOnβ€˜πΊ) = βˆ…)
12 fvprc 6883 . . . . 5 (Β¬ 𝐺 ∈ V β†’ (Vtxβ€˜πΊ) = βˆ…)
1312orcd 872 . . . 4 (Β¬ 𝐺 ∈ V β†’ ((Vtxβ€˜πΊ) = βˆ… ∨ β„•0 = βˆ…))
14 0mpo0 7497 . . . 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 2770 . 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 846   = wceq 1534   ∈ wcel 2099  {crab 3427  Vcvv 3469  βˆ…c0 4318  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  0cc0 11130  β„•0cn0 12494  Vtxcvtx 28796   ClWWalksN cclwwlkn 29821  ClWWalksNOncclwwlknon 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-1cn 11188  ax-addcl 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12235  df-n0 12495  df-clwwlknon 29885
This theorem is referenced by:  clwwlknon  29887  clwwlk0on0  29889  clwwlknon0  29890  2clwwlk2clwwlklem  30143
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