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Theorem wwlksnextsur 26795

Description: Lemma for wwlksnextbij 26797. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)

**Hypotheses**
Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
wwlksnextbij.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ ( lastS ‘𝑡))

**Assertion**
Ref	Expression
wwlksnextsur	⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑤,𝑊 𝑡,𝐷 𝑛,𝐸,𝑤 𝑡,𝑁,𝑤 𝑡,𝑅 𝑛,𝑉,𝑤 𝑛,𝑊 𝑡,𝑛,𝑁,𝑤 Allowed substitution hints: 𝐷(𝑤,𝑛) 𝑅(𝑤,𝑛) 𝐸(𝑡) 𝐹(𝑤,𝑡,𝑛) 𝐺(𝑡,𝑛) 𝑉(𝑡) 𝑊(𝑡)

Proof of Theorem wwlksnextsur Dummy variables 𝑖 𝑑 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnextbij0.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	wwlknbp 26733	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉))
3		simp2 1062	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉) → 𝑁 ∈ ℕ₀)
4		wwlksnextbij0.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
5		wwlksnextbij0.d	. . . 4 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
6		wwlksnextbij.r	. . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
7		wwlksnextbij.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ ( lastS ‘𝑡))
8	1, 4, 5, 6, 7	wwlksnextfun 26793	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷⟶𝑅)
9	2, 3, 8	3syl 18	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷⟶𝑅)
10		preq2 4269	. . . . . 6 ⊢ (𝑛 = 𝑟 → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), 𝑟})
11	10	eleq1d 2686	. . . . 5 ⊢ (𝑛 = 𝑟 → ({( lastS ‘𝑊), 𝑛} ∈ 𝐸 ↔ {( lastS ‘𝑊), 𝑟} ∈ 𝐸))
12	11, 6	elrab2 3366	. . . 4 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸))
13	1, 4	wwlksnext 26788	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
14	13	3expb 1266	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
15		s1cl 13382	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ 𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
16		swrdccat1 13457	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
17	15, 16	sylan2 491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
18	17	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
19	18	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
20		opeq2 4403	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 + 1) = (#‘𝑊) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑊)⟩)
21	20	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑊) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑊)⟩)
22	21	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩))
23	22	eqeq1d 2624	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
24	23	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
25	19, 24	sylibrd 249	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
26	25	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
27	1, 4	wwlknp 26734	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
28	26, 27	syl11 33	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
29	28	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
30	29	impcom 446	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊)
31		lswccats1 13411	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
32	31	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
33	32	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
34	33	3ad2ant3 1084	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉) → (𝑟 ∈ 𝑉 → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
35	2, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑟 ∈ 𝑉 → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
36	35	imp 445	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
37	36	preq2d 4275	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → {( lastS ‘𝑊), 𝑟} = {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))})
38	37	eleq1d 2686	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({( lastS ‘𝑊), 𝑟} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
39	38	biimpd 219	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({( lastS ‘𝑊), 𝑟} ∈ 𝐸 → {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
40	39	impr 649	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
41	14, 30, 40	jca32 558	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
42	34, 2	syl11 33	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
43	42	adantr 481	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
44	43	impcom 446	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
45		ovexd 6680	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
46		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
47		oveq1 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩))
48	47	eqeq1d 2624	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
49		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ( lastS ‘𝑑) = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
50	49	preq2d 4275	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {( lastS ‘𝑊), ( lastS ‘𝑑)} = {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))})
51	50	eleq1d 2686	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
52	48, 51	anbi12d 747	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸) ↔ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
53	46, 52	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ↔ ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))))
54	49	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = ( lastS ‘𝑑) ↔ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
55	53, 54	anbi12d 747	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)) ↔ (((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))))
56	55	bicomd 213	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
57	56	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
58	57	biimpd 219	. . . . . . . . . 10 ⊢ (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) → ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
59	45, 58	spcimedv 3292	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑))))
60	41, 44, 59	mp2and 715	. . . . . . . 8 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)))
61		oveq1 6657	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
62	61	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
63		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑑 → ( lastS ‘𝑤) = ( lastS ‘𝑑))
64	63	preq2d 4275	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑑)})
65	64	eleq1d 2686	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸))
66	62, 65	anbi12d 747	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑑 → (((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)))
67	66	elrab 3363	. . . . . . . . . 10 ⊢ (𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ↔ (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)))
68	67	anbi1i 731	. . . . . . . . 9 ⊢ ((𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)))
69	68	exbii 1774	. . . . . . . 8 ⊢ (∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)) ↔ ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = ( lastS ‘𝑑)))
70	60, 69	sylibr 224	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)))
71		df-rex 2918	. . . . . . 7 ⊢ (∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}𝑟 = ( lastS ‘𝑑) ↔ ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = ( lastS ‘𝑑)))
72	70, 71	sylibr 224	. . . . . 6 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}𝑟 = ( lastS ‘𝑑))
73	1, 4, 5	wwlksnextwrd 26792	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
74	73	adantr 481	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
75	74	rexeqdv 3145	. . . . . 6 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (∃𝑑 ∈ 𝐷 𝑟 = ( lastS ‘𝑑) ↔ ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}𝑟 = ( lastS ‘𝑑)))
76	72, 75	mpbird 247	. . . . 5 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = ( lastS ‘𝑑))
77		fveq2 6191	. . . . . . . 8 ⊢ (𝑡 = 𝑑 → ( lastS ‘𝑡) = ( lastS ‘𝑑))
78		fvex 6201	. . . . . . . 8 ⊢ ( lastS ‘𝑑) ∈ V
79	77, 7, 78	fvmpt 6282	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = ( lastS ‘𝑑))
80	79	eqeq2d 2632	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝑟 = (𝐹‘𝑑) ↔ 𝑟 = ( lastS ‘𝑑)))
81	80	rexbiia 3040	. . . . 5 ⊢ (∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑) ↔ ∃𝑑 ∈ 𝐷 𝑟 = ( lastS ‘𝑑))
82	76, 81	sylibr 224	. . . 4 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
83	12, 82	sylan2b 492	. . 3 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑅) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
84	83	ralrimiva 2966	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
85		dffo3 6374	. 2 ⊢ (𝐹:𝐷–onto→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑)))
86	9, 84, 85	sylanbrc 698	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 {cpr 4179 ⟨cop 4183 ↦ cmpt 4729 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 2c2 11070 ℕ₀cn0 11292 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 ++ cconcat 13293 ⟨“cs1 13294 substr csubstr 13295 Vtxcvtx 25874 Edgcedg 25939 WWalksN cwwlksn 26718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-wwlks 26722 df-wwlksn 26723

This theorem is referenced by: wwlksnextbij0 26796

W3C validator