Theorem wwlksn 30345

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 6874	. . . . . 6 ⊢ (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺))
2	1	adantl 487	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺))
3		oveq1 7416	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1))
4	3	eqeq2d 2771	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
5	4	adantr 486	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1)))
6	2, 5	rabeqbidv 3429	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
7		df-wwlksn 30339	. . . 4 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
8		fvex 6887	. . . . 5 ⊢ (WWalks‘𝐺) ∈ V
9	8	rabex 5300	. . . 4 ⊢ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V
10	6, 7, 9	ovmpoa 7564	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
11	10	expcom 419	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}))
12	7	reldmmpo 7543	. . . . 5 ⊢ Rel dom WWalksN
13	12	ovprc2 7449	. . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅)
14		fvprc 6866	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (WWalks‘𝐺) = ∅)
15	14	rabeqdv 3427	. . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)})
16		rab0 4335	. . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅
17	15, 16	eqtrdi 2811	. . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅)
18	13, 17	eqtr4d 2798	. . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
19	18	a1d 26	. 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)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412  Vcvv 3450  ∅c0 4279  ‘cfv 6528  (class class class)co 7409  1c1 11158   + caddc 11160  ℕ₀cn0 12561  ♯chash 14427  WWalkscwwlks 30333   WWalksN cwwlksn 30334

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6484  df-fun 6530  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-wwlksn 30339

This theorem is used by:  iswwlksn  30346  wwlksn0s  30369  0enwwlksnge1  30372  wlknwwlksnbij  30396  wwlksnfi  30414