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Theorem clwwlknonmpo 27881
 Description: (ClWWalksNOn‘𝐺) 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 (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
Distinct variable group:   𝑛,𝐺,𝑣,𝑤

Proof of Theorem clwwlknonmpo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6645 . . . 4 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2 eqidd 2799 . . . 4 (𝑔 = 𝐺 → ℕ0 = ℕ0)
3 oveq2 7143 . . . . 5 (𝑔 = 𝐺 → (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
43rabeqdv 3432 . . . 4 (𝑔 = 𝐺 → {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
51, 2, 4mpoeq123dv 7208 . . 3 (𝑔 = 𝐺 → (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
6 df-clwwlknon 27880 . . 3 ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
7 fvex 6658 . . . 4 (Vtx‘𝐺) ∈ V
8 nn0ex 11893 . . . 4 0 ∈ V
97, 8mpoex 7762 . . 3 (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V
105, 6, 9fvmpt 6745 . 2 (𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
11 fvprc 6638 . . 3 𝐺 ∈ V → (ClWWalksNOn‘𝐺) = ∅)
12 fvprc 6638 . . . . 5 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
1312orcd 870 . . . 4 𝐺 ∈ V → ((Vtx‘𝐺) = ∅ ∨ ℕ0 = ∅))
14 0mpo0 7216 . . . 4 (((Vtx‘𝐺) = ∅ ∨ ℕ0 = ∅) → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = ∅)
1513, 14syl 17 . . 3 𝐺 ∈ V → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = ∅)
1611, 15eqtr4d 2836 . 2 𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
1710, 16pm2.61i 185 1 (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  ∅c0 4243  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  0cc0 10528  ℕ0cn0 11887  Vtxcvtx 26796   ClWWalksN cclwwlkn 27816  ClWWalksNOncclwwlknon 27879 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-1cn 10586  ax-addcl 10588 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-nn 11628  df-n0 11888  df-clwwlknon 27880 This theorem is referenced by:  clwwlknon  27882  clwwlk0on0  27884  clwwlknon0  27885  2clwwlk2clwwlklem  28138
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