Theorem wwlknfi 26266

Description: The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.)

Assertion

Ref	Expression
wwlknfi	⊢ (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)

Proof of Theorem wwlknfi

Dummy variables 𝑖 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlkn 26210	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)})
2		df-rab 2905	. . . . . . 7 ⊢ {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)} = {𝑤 ∣ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))}
3	2	a1i 11	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)} = {𝑤 ∣ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))})
4		iswwlk 26211	. . . . . . . . . 10 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
5	4	3adant3 1074	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
6	5	anbi1d 737	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
7	6	abbidv 2728	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑤 ∣ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} = {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))})
8		3anan12 1044	. . . . . . . . . . 11 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
9	8	anbi1i 727	. . . . . . . . . 10 ⊢ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ ((𝑤 ∈ Word 𝑉 ∧ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ (#‘𝑤) = (𝑁 + 1)))
10		anass 679	. . . . . . . . . 10 ⊢ (((𝑤 ∈ Word 𝑉 ∧ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
11	9, 10	bitri 263	. . . . . . . . 9 ⊢ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))))
12	11	abbii 2726	. . . . . . . 8 ⊢ {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)))}
13		df-rab 2905	. . . . . . . 8 ⊢ {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)))}
14	12, 13	eqtr4i 2635	. . . . . . 7 ⊢ {𝑤 ∣ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} = {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))}
15	7, 14	syl6eq 2660	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑤 ∣ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} = {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))})
16	1, 3, 15	3eqtrd 2648	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))})
17	16	adantr 480	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑉 ∈ Fin) → ((𝑉 WWalksN 𝐸)‘𝑁) = {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))})
18		peano2nn0 11210	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
19	18	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑁 + 1) ∈ ℕ0)
20	19	anim2i 591	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0)) → (𝑉 ∈ Fin ∧ (𝑁 + 1) ∈ ℕ0))
21	20	ancoms 468	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑉 ∈ Fin) → (𝑉 ∈ Fin ∧ (𝑁 + 1) ∈ ℕ0))
22		wrdnfi 13193	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ (𝑁 + 1) ∈ ℕ0) → {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = (𝑁 + 1)} ∈ Fin)
23	21, 22	syl 17	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑉 ∈ Fin) → {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = (𝑁 + 1)} ∈ Fin)
24		simpr 476	. . . . . . 7 ⊢ (((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) → (#‘𝑤) = (𝑁 + 1))
25	24	rgenw 2908	. . . . . 6 ⊢ ∀𝑤 ∈ Word 𝑉(((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) → (#‘𝑤) = (𝑁 + 1))
26		ss2rab 3641	. . . . . 6 ⊢ ({𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} ⊆ {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = (𝑁 + 1)} ↔ ∀𝑤 ∈ Word 𝑉(((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1)) → (#‘𝑤) = (𝑁 + 1)))
27	25, 26	mpbir 220	. . . . 5 ⊢ {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} ⊆ {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = (𝑁 + 1)}
28		ssfi 8065	. . . . 5 ⊢ (({𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = (𝑁 + 1)} ∈ Fin ∧ {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} ⊆ {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = (𝑁 + 1)}) → {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} ∈ Fin)
29	23, 27, 28	sylancl 693	. . . 4 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑉 ∈ Fin) → {𝑤 ∈ Word 𝑉 ∣ ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (𝑁 + 1))} ∈ Fin)
30	17, 29	eqeltrd 2688	. . 3 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ 𝑉 ∈ Fin) → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
31	30	ex 449	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin))
32		wwlknndef 26265	. . . . 5 ⊢ ((𝑉 ∉ V ∨ 𝐸 ∉ V ∨ 𝑁 ∉ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = ∅)
33		3ioran 1049	. . . . . 6 ⊢ (¬ (𝑉 ∉ V ∨ 𝐸 ∉ V ∨ 𝑁 ∉ ℕ0) ↔ (¬ 𝑉 ∉ V ∧ ¬ 𝐸 ∉ V ∧ ¬ 𝑁 ∉ ℕ0))
34		nnel 2892	. . . . . . 7 ⊢ (¬ 𝑉 ∉ V ↔ 𝑉 ∈ V)
35		nnel 2892	. . . . . . 7 ⊢ (¬ 𝐸 ∉ V ↔ 𝐸 ∈ V)
36		nnel 2892	. . . . . . 7 ⊢ (¬ 𝑁 ∉ ℕ0 ↔ 𝑁 ∈ ℕ0)
37	34, 35, 36	3anbi123i 1244	. . . . . 6 ⊢ ((¬ 𝑉 ∉ V ∧ ¬ 𝐸 ∉ V ∧ ¬ 𝑁 ∉ ℕ0) ↔ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
38	33, 37	sylbb 208	. . . . 5 ⊢ (¬ (𝑉 ∉ V ∨ 𝐸 ∉ V ∨ 𝑁 ∉ ℕ0) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
39	32, 38	nsyl4 155	. . . 4 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = ∅)
40		0fin 8073	. . . . 5 ⊢ ∅ ∈ Fin
41	40	a1i 11	. . . 4 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ∅ ∈ Fin)
42	39, 41	eqeltrd 2688	. . 3 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)
43	42	a1d 25	. 2 ⊢ (¬ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin))
44	31, 43	pm2.61i 175	1 ⊢ (𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   WWalks cwwlk 26205   WWalksN cwwlkn 26206

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wlknfi  26267  hashwwlkext  26274  rusgranumwlks  26483  clwlknclwlkdifnum  26488