Theorem av-numclwwlkovq 41529

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). (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 30-May-2021.)

Hypotheses

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

Assertion

Ref	Expression
av-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 av-numclwwlkovq

Step	Hyp	Ref	Expression
1		oveq1 6556	. . . 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 2621	. . . . 5 ⊢ (𝑣 = 𝑋 → ((𝑤‘0) = 𝑣 ↔ (𝑤‘0) = 𝑋))
4		neeq2 2845	. . . . 5 ⊢ (𝑣 = 𝑋 → (( lastS ‘𝑤) ≠ 𝑣 ↔ ( lastS ‘𝑤) ≠ 𝑋))
5	3, 4	anbi12d 743	. . . 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 3168	. 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		av-numclwwlk.q	. 2 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})
9		ovex 6577	. . 3 ⊢ (𝑁 WWalkSN 𝐺) ∈ V
10	9	rabex 4740	. 2 ⊢ {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)} ∈ V
11	7, 8, 10	ovmpt2a 6689	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  ℕcn 10897   lastS clsw 13147  Vtxcvtx 25673   WWalkSN cwwlksn 41029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  av-numclwwlkqhash  41530  av-numclwwlk2lem1  41532  av-numclwlk2lem2f  41533  av-numclwlk2lem2f1o  41535