Theorem wwlksn 27617

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		fveq2 6672	. . . . . 6 ⊢ (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
2	1	adantl 484	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
3		oveq1 7165	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
4	3	eqeq2d 2834	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
5	4	adantr 483	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
6	2, 5	rabeqbidv 3487	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
7		df-wwlksn 27611	. . . 4 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
8		fvex 6685	. . . . 5 ⊢ (WWalks‘𝐺) ∈ V
9	8	rabex 5237	. . . 4 ⊢ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V
10	6, 7, 9	ovmpoa 7307	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
11	10	expcom 416	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
12	7	reldmmpo 7287	. . . . 5 ⊢ Rel dom WWalksN
13	12	ovprc2 7198	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
14		fvprc 6665	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
15	14	rabeqdv 3486	. . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)})
16		rab0 4339	. . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅
17	15, 16	syl6eq 2874	. . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅)
18	13, 17	eqtr4d 2861	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
19	18	a1d 25	. 2 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
20	11, 19	pm2.61i 184	1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144  Vcvv 3496  ∅c0 4293  ‘cfv 6357  (class class class)co 7158  1c1 10540   + caddc 10542  ℕ₀cn0 11900  ♯chash 13693  WWalkscwwlks 27605   WWalksN cwwlksn 27606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-wwlksn 27611

This theorem is referenced by:  iswwlksn  27618  wwlksn0s  27641  0enwwlksnge1  27644  wlknwwlksnbij  27668  wwlksnfi  27686  wwlksnfiOLD  27687