Theorem clwwlknonmpo 30695

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.

Step	Hyp	Ref	Expression
1		fveq2 6896	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
2		eqidd 2764	. . . 4 ⊢ (𝑔 = 𝐺 → ℕ0 = ℕ0)
3		oveq2 7438	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑛 ClWWalksN 𝑔) = (𝑛 ClWWalksN 𝐺))
4	3	rabeqdv 3431	. . . 4 ⊢ (𝑔 = 𝐺 → {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣} = {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
5	1, 2, 4	mpoeq123dv 7505	. . 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 30694	. . 3 ⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
7		fvex 6909	. . . 4 ⊢ (Vtx‘𝐺) ∈ V
8		nn0ex 12625	. . . 4 ⊢ ℕ0 ∈ V
9	7, 8	mpoex 8103	. . 3 ⊢ (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) ∈ V
10	5, 6, 9	fvmpt 7004	. 2 ⊢ (𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
11		fvprc 6888	. . 3 ⊢ (¬ 𝐺 ∈ V → (ClWWalksNOn‘𝐺) = ∅)
12		fvprc 6888	. . . . 5 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
13	12	orcd 887	. . . 4 ⊢ (¬ 𝐺 ∈ V → ((Vtx‘𝐺) = ∅ ∨ ℕ0 = ∅))
14		0mpo0 7513	. . . 4 ⊢ (((Vtx‘𝐺) = ∅ ∨ ℕ0 = ∅) → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = ∅)
15	13, 14	syl 18	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}) = ∅)
16	11, 15	eqtr4d 2801	. 2 ⊢ (¬ 𝐺 ∈ V → (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))
17	10, 16	pm2.61i 184	1 ⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1569   ∈ wcel 2146  {crab 3416  Vcvv 3454  ∅c0 4281  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  0cc0 11213  ℕ₀cn0 12619  Vtxcvtx 29589   ClWWalksN cclwwlkn 30630  ClWWalksNOncclwwlknon 30693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-1cn 11271  ax-addcl 11273

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-nn 12349  df-n0 12620  df-clwwlknon 30694

This theorem is used by:  clwwlknon  30696  clwwlk0on0  30698  clwwlknon0  30699  2clwwlk2clwwlklem  30962