Theorem numclwwlkovq 30158

Description: Value of operation 𝑄, mapping a vertex 𝑣 and a positive integer 𝑛 to the not closed walks v(0) ... v(n) of length 𝑛 from a fixed vertex 𝑣 = v(0). "Not closed" means v(n) =/= v(0). Remark: 𝑛 ∈ ℕ₀ would not be useful: numclwwlkqhash 30159 would not hold, because (𝐾↑0) = 1! (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 30-May-2021.)

Hypotheses

Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})

Assertion

Ref	Expression
numclwwlkovq	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤

Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝑉(𝑤)

Proof of Theorem numclwwlkovq

Step	Hyp	Ref	Expression
1		oveq1 7421	. . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
2	1	adantl 481	. . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (𝑛 WWalksN 𝐺) = (𝑁 WWalksN 𝐺))
3		eqeq2 2739	. . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
4		neeq2 2999	. . . . 5 ⊢ (𝑣 = 𝑋 → ((lastS‘𝑤) ≠ 𝑣 ↔ (lastS‘𝑤) ≠ 𝑋))
5	3, 4	anbi12d 630	. . . 4 ⊢ (𝑣 = 𝑋 → (((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)))
6	5	adantr 480	. . 3 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → (((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣) ↔ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)))
7	2, 6	rabeqbidv 3444	. 2 ⊢ ((𝑣 = 𝑋 ∧ 𝑛 = 𝑁) → {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})
8		numclwwlk.q	. 2 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
9		ovex 7447	. . 3 ⊢ (𝑁 WWalksN 𝐺) ∈ V
10	9	rabex 5328	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ∈ V
11	7, 8, 10	ovmpoa 7568	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  {crab 3427  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  0cc0 11124  ℕcn 12228  lastSclsw 14530  Vtxcvtx 28783   WWalksN cwwlksn 29611

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-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  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-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419

This theorem is referenced by:  numclwwlkqhash  30159  numclwwlk2lem1  30160  numclwlk2lem2f  30161  numclwlk2lem2f1o  30163