Theorem rusgranumwlkl1 26474

Description: In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.)

Assertion

Ref	Expression
rusgranumwlkl1	⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = 𝐾)

Distinct variable groups:   𝑤,𝐸   𝑤,𝐾   𝑤,𝑃   𝑤,𝑉

Proof of Theorem rusgranumwlkl1

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rusisusgra 26458	. . . . . . . . . 10 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → 𝑉 USGrph 𝐸)
2		usgrav 25867	. . . . . . . . . 10 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3	1, 2	syl 17	. . . . . . . . 9 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4		1nn0 11185	. . . . . . . . . 10 ⊢ 1 ∈ ℕ0
5	4	a1i 11	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝑉 → 1 ∈ ℕ0)
6	3, 5	anim12i 588	. . . . . . . 8 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 1 ∈ ℕ0))
7		df-3an 1033	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 1 ∈ ℕ0))
8	6, 7	sylibr 223	. . . . . . 7 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0))
9		iswwlkn 26212	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ↔ (𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (1 + 1))))
10		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)))
11	10	3adant3 1074	. . . . . . . . 9 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → (𝑤 ∈ (𝑉 WWalks 𝐸) ↔ (𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
12	11	anbi1d 737	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → ((𝑤 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1))))
13	9, 12	bitrd 267	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 1 ∈ ℕ0) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1))))
14	8, 13	syl 17	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1))))
15	14	anbi1d 737	. . . . 5 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃)))
16		1p1e2 11011	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
17	16	eqeq2i 2622	. . . . . . . . . 10 ⊢ ((#‘𝑤) = (1 + 1) ↔ (#‘𝑤) = 2)
18	17	a1i 11	. . . . . . . . 9 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((#‘𝑤) = (1 + 1) ↔ (#‘𝑤) = 2))
19	18	anbi2d 736	. . . . . . . 8 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = 2)))
20		3anass 1035	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)))
21	20	a1i 11	. . . . . . . . . . 11 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))))
22		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
23		hash0 13019	. . . . . . . . . . . . . . . 16 ⊢ (#‘∅) = 0
24	22, 23	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → (#‘𝑤) = 0)
25		2ne0 10990	. . . . . . . . . . . . . . . . 17 ⊢ 2 ≠ 0
26	25	nesymi 2839	. . . . . . . . . . . . . . . 16 ⊢ ¬ 0 = 2
27		eqeq1 2614	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑤) = 0 → ((#‘𝑤) = 2 ↔ 0 = 2))
28	26, 27	mtbiri 316	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑤) = 0 → ¬ (#‘𝑤) = 2)
29	24, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∅ → ¬ (#‘𝑤) = 2)
30	29	necon2ai 2811	. . . . . . . . . . . . 13 ⊢ ((#‘𝑤) = 2 → 𝑤 ≠ ∅)
31	30	adantl 481	. . . . . . . . . . . 12 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → 𝑤 ≠ ∅)
32	31	biantrurd 528	. . . . . . . . . . 11 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))))
33		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑤) = 2 → ((#‘𝑤) − 1) = (2 − 1))
34		2m1e1 11012	. . . . . . . . . . . . . . . . 17 ⊢ (2 − 1) = 1
35	33, 34	syl6eq 2660	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑤) = 2 → ((#‘𝑤) − 1) = 1)
36	35	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑤) = 2 → (0..^((#‘𝑤) − 1)) = (0..^1))
37	36	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → (0..^((#‘𝑤) − 1)) = (0..^1))
38	37	raleqdv 3121	. . . . . . . . . . . . 13 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^1){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸))
39		fzo01 12417	. . . . . . . . . . . . . . 15 ⊢ (0..^1) = {0}
40	39	raleqi 3119	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ (0..^1){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ {0} {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)
41		c0ex 9913	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
42		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑤‘𝑖) = (𝑤‘0))
43		oveq1 6556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
44		0p1e1 11009	. . . . . . . . . . . . . . . . . . 19 ⊢ (0 + 1) = 1
45	43, 44	syl6eq 2660	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 0 → (𝑖 + 1) = 1)
46	45	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑤‘(𝑖 + 1)) = (𝑤‘1))
47	42, 46	preq12d 4220	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} = {(𝑤‘0), (𝑤‘1)})
48	47	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
49	41, 48	ralsn 4169	. . . . . . . . . . . . . 14 ⊢ (∀𝑖 ∈ {0} {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)
50	40, 49	bitri 263	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ (0..^1){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)
51	38, 50	syl6bb 275	. . . . . . . . . . . 12 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
52	51	anbi2d 736	. . . . . . . . . . 11 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
53	21, 32, 52	3bitr2d 295	. . . . . . . . . 10 ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) ∧ (#‘𝑤) = 2) → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
54	53	ex 449	. . . . . . . . 9 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((#‘𝑤) = 2 → ((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))))
55	54	pm5.32rd 670	. . . . . . . 8 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = 2) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2)))
56	19, 55	bitrd 267	. . . . . . 7 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2)))
57	56	anbi1d 737	. . . . . 6 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃)))
58		anass 679	. . . . . 6 ⊢ ((((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ (#‘𝑤) = 2) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)))
59	57, 58	syl6bb 275	. . . . 5 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((((𝑤 ≠ ∅ ∧ 𝑤 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘𝑤) = (1 + 1)) ∧ (𝑤‘0) = 𝑃) ↔ ((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
60		anass 679	. . . . . . 7 ⊢ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))))
61		ancom 465	. . . . . . . . 9 ⊢ (({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
62		df-3an 1033	. . . . . . . . 9 ⊢ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ↔ (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃) ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
63	61, 62	bitr4i 266	. . . . . . . 8 ⊢ (({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))
64	63	anbi2i 726	. . . . . . 7 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ({(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃))) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
65	60, 64	bitri 263	. . . . . 6 ⊢ (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)))
66	65	a1i 11	. . . . 5 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (((𝑤 ∈ Word 𝑉 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸) ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃)) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))))
67	15, 59, 66	3bitrd 293	. . . 4 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → ((𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∧ (𝑤‘0) = 𝑃) ↔ (𝑤 ∈ Word 𝑉 ∧ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸))))
68	67	rabbidva2 3162	. . 3 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)})
69	68	fveq2d 6107	. 2 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}))
70		rusgranumwwlkl1 26473	. 2 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾)
71	69, 70	eqtrd 2644	1 ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = 𝐾)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  2c2 10947  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   WWalks cwwlk 26205   WWalksN cwwlkn 26206   RegUSGrph crusgra 26450

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-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-nbgra 25949  df-wwlk 26207  df-wwlkn 26208  df-vdgr 26421  df-rgra 26451  df-rusgra 26452

This theorem is referenced by:  rusgranumwlkb1  26481