Theorem numclwwlkovq 30459

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 30460 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 7367	. . . 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 2749	. . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
4		neeq2 2996	. . . . 5 ⊢ (𝑣 = 𝑋 → ((lastS‘𝑤) ≠ 𝑣 ↔ (lastS‘𝑤) ≠ 𝑋))
5	3, 4	anbi12d 633	. . . 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 3408	. 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 7393	. . 3 ⊢ (𝑁 WWalksN 𝐺) ∈ V
10	9	rabex 5276	. 2 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} ∈ V
11	7, 8, 10	ovmpoa 7515	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362  0cc0 11029  ℕcn 12165  lastSclsw 14515  Vtxcvtx 29079   WWalksN cwwlksn 29909

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  numclwwlkqhash  30460  numclwwlk2lem1  30461  numclwlk2lem2f  30462  numclwlk2lem2f1o  30464