Theorem wwlksn 41040

Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)

Assertion

Ref	Expression
wwlksn	⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁

Proof of Theorem wwlksn

Dummy variables 𝑔 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wwlksn 41034	. . . . 5 ⊢ WWalkSN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
2	1	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → WWalkSN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)}))
3		fveq2 6103	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (WWalkS‘𝑔) = (WWalkS‘𝐺))
4	3	adantl 481	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (WWalkS‘𝑔) = (WWalkS‘𝐺))
5		oveq1 6556	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
6	5	eqeq2d 2620	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((#‘𝑤) = (𝑛 + 1) ↔ (#‘𝑤) = (𝑁 + 1)))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((#‘𝑤) = (𝑛 + 1) ↔ (#‘𝑤) = (𝑁 + 1)))
8	4, 7	rabeqbidv 3168	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
9	8	adantl 481	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
10		simpl 472	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → 𝑁 ∈ ℕ0)
11		simpr 476	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → 𝐺 ∈ V)
12		fvex 6113	. . . . . 6 ⊢ (WWalkS‘𝐺) ∈ V
13	12	rabex 4740	. . . . 5 ⊢ {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} ∈ V
14	13	a1i 11	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} ∈ V)
15	2, 9, 10, 11, 14	ovmpt2d 6686	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
16	15	expcom 450	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)}))
17	1	reldmmpt2 6669	. . . . 5 ⊢ Rel dom WWalkSN
18	17	ovprc2 6583	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalkSN 𝐺) = ∅)
19		fvprc 6097	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (WWalkS‘𝐺) = ∅)
20	19	rabeqdv 3167	. . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (#‘𝑤) = (𝑁 + 1)})
21		rab0 3909	. . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (#‘𝑤) = (𝑁 + 1)} = ∅
22	20, 21	syl6eq 2660	. . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} = ∅)
23	18, 22	eqtr4d 2647	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})
24	23	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)}))
25	16, 24	pm2.61i 175	1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1c1 9816   + caddc 9818  ℕ₀cn0 11169  #chash 12979  WWalkScwwlks 41028   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-8 1979  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-pow 4769  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-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  df-wwlksn 41034

This theorem is referenced by:  iswwlksn  41041  wwlksn0s  41057  0enwwlksnge1  41060  wwlksnfi  41112