wwlksnextsur - Mathbox for Alexander van der Vekens

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Theorem wwlksnextsur 41106

Description: Lemma for wwlksnextbij 41108. (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 41044	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉))
3		simp2 1055	. . 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 41104	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷⟶𝑅)
9	2, 3, 8	3syl 18	. 2 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐹:𝐷⟶𝑅)
10		preq2 4213	. . . . . 6 ⊢ (𝑛 = 𝑟 → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), 𝑟})
11	10	eleq1d 2672	. . . . 5 ⊢ (𝑛 = 𝑟 → ({( lastS ‘𝑊), 𝑛} ∈ 𝐸 ↔ {( lastS ‘𝑊), 𝑟} ∈ 𝐸))
12	11, 6	elrab2 3333	. . . 4 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸))
13	1, 4	wwlksnext 41099	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺))
14	13	3expb 1258	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺))
15	1, 4	wwlknp 41045	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
16		s1cl 13235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ 𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
17		swrdccat1 13309	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
18	16, 17	sylan2 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
19	18	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
20	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
21		opeq2 4341	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 + 1) = (#‘𝑊) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑊)⟩)
22	21	eqcoms 2618	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑊) = (𝑁 + 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, (#‘𝑊)⟩)
23	22	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩))
24	23	eqeq1d 2612	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
25	24	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊))
26	20, 25	sylibrd 248	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
27	26	3adant3 1074	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
28	15, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
29	28	com12 32	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
30	29	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
31	30	impcom 445	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊)
32		lswccats1 13263	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
33	32	eqcomd 2616	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
34	33	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
35	34	3ad2ant3 1077	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ Word 𝑉) → (𝑟 ∈ 𝑉 → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
36	2, 35	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑟 ∈ 𝑉 → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
37	36	imp 444	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟 ∈ 𝑉) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
38	37	preq2d 4219	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟 ∈ 𝑉) → {( lastS ‘𝑊), 𝑟} = {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))})
39	38	eleq1d 2672	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({( lastS ‘𝑊), 𝑟} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
40	39	biimpd 218	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({( lastS ‘𝑊), 𝑟} ∈ 𝐸 → {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
41	40	impr 647	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
42	14, 31, 41	jca32 556	. . . . . . . . 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)))
43	36	com12 32	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
45	44	impcom 445	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
46		ovex 6577	. . . . . . . . . . 11 ⊢ (𝑊 ++ ⟨“𝑟”⟩) ∈ V
47	46	a1i 11	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
48		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalkSN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalkSN 𝐺)))
49		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩))
50	49	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
51		fveq2 6103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ( lastS ‘𝑑) = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩)))
52	51	preq2d 4219	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {( lastS ‘𝑊), ( lastS ‘𝑑)} = {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))})
53	52	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
54	50, 53	anbi12d 743	. . . . . . . . . . . . . . 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)))
55	48, 54	anbi12d 743	. . . . . . . . . . . . . 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))))
56	51	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = ( lastS ‘𝑑) ↔ 𝑟 = ( lastS ‘(𝑊 ++ ⟨“𝑟”⟩))))
57	55, 56	anbi12d 743	. . . . . . . . . . . . 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”⟩)))))
58	57	bicomd 212	. . . . . . . . . . . 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))))
59	58	adantl 481	. . . . . . . . . . 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))))
60	59	biimpd 218	. . . . . . . . . 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))))
61	47, 60	spcimedv 3265	. . . . . . . . 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))))
62	42, 45, 61	mp2and 711	. . . . . . . 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)))
63		oveq1 6556	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
64	63	eqeq1d 2612	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
65		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑑 → ( lastS ‘𝑤) = ( lastS ‘𝑑))
66	65	preq2d 4219	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑑)})
67	66	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑑)} ∈ 𝐸))
68	64, 67	anbi12d 743	. . . . . . . . . . 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)))
69	68	elrab 3331	. . . . . . . . . 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)))
70	69	anbi1i 727	. . . . . . . . 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)))
71	70	exbii 1764	. . . . . . . 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)))
72	62, 71	sylibr 223	. . . . . . 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)))
73		df-rex 2902	. . . . . . 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)))
74	72, 73	sylibr 223	. . . . . 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))
75	1, 4, 5	wwlksnextwrd 41103	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
76	75	adantr 480	. . . . . . 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)})
77	76	rexeqdv 3122	. . . . . 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)))
78	74, 77	mpbird 246	. . . . 5 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = ( lastS ‘𝑑))
79		fveq2 6103	. . . . . . . 8 ⊢ (𝑡 = 𝑑 → ( lastS ‘𝑡) = ( lastS ‘𝑑))
80		fvex 6113	. . . . . . . 8 ⊢ ( lastS ‘𝑑) ∈ V
81	79, 7, 80	fvmpt 6191	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = ( lastS ‘𝑑))
82	81	eqeq2d 2620	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝑟 = (𝐹‘𝑑) ↔ 𝑟 = ( lastS ‘𝑑)))
83	82	rexbiia 3022	. . . . 5 ⊢ (∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑) ↔ ∃𝑑 ∈ 𝐷 𝑟 = ( lastS ‘𝑑))
84	78, 83	sylibr 223	. . . 4 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {( lastS ‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
85	12, 84	sylan2b 491	. . 3 ⊢ ((𝑊 ∈ (𝑁 WWalkSN 𝐺) ∧ 𝑟 ∈ 𝑅) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
86	85	ralrimiva 2949	. 2 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
87		dffo3 6282	. 2 ⊢ (𝐹:𝐷–onto→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑)))
88	9, 86, 87	sylanbrc 695	1 ⊢ (𝑊 ∈ (𝑁 WWalkSN 𝐺) → 𝐹:𝐷–onto→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 {cpr 4127 ⟨cop 4131 ↦ cmpt 4643 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 2c2 10947 ℕ₀cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 ++ cconcat 13148 ⟨“cs1 13149 substr csubstr 13150 Vtxcvtx 25673 Edgcedga 25792 WWalkSN cwwlksn 41029

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-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-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-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-rp 11709 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-wwlks 41033 df-wwlksn 41034

This theorem is referenced by: wwlksnextbij0 41107

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