Theorem wwlksnextsurj 30190

Description: Lemma for wwlksnextbij 30192. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij0.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))

Assertion

Ref	Expression
wwlksnextsurj	⊢ (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 wwlksnextsurj

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 30132	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
3		simp2 1153	. . 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 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
6		wwlksnextbij0.r	. . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
7		wwlksnextbij0.f	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))
8	1, 4, 5, 6, 7	wwlksnextfun 30188	. . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
9	2, 3, 8	3syl 19	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷⟶𝑅)
10		preq2 4705	. . . . . 6 ⊢ (𝑛 = 𝑟 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑟})
11	10	eleq1d 2854	. . . . 5 ⊢ (𝑛 = 𝑟 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
12	11, 6	elrab2 3663	. . . 4 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
13	1, 4	wwlksnext 30183	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
14	13	3expb 1136	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
15		s1cl 14640	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ 𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
16		pfxccat1 14739	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
17	15, 16	sylan2 604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
18	17	ex 417	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
19	18	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
20		oveq2 7419	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 + 1) = (♯‘𝑊) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
21	20	eqcoms 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
22	21	eqeq1d 2771	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
23	22	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
24	19, 23	sylibrd 262	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
25	24	3adant3 1148	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
26	1, 4	wwlknp 30133	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
27	25, 26	syl11 34	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
28	27	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
29	28	impcom 412	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊)
30		lswccats1 14672	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → (lastS‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
31	30	eqcomd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
32	31	ex 417	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
33	32	3ad2ant3 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
34	2, 33	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
35	34	imp 411	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
36	35	preq2d 4711	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → {(lastS‘𝑊), 𝑟} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
37	36	eleq1d 2854	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
38	37	biimpd 232	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
39	38	impr 459	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
40	14, 29, 39	jca32 524	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
41	33, 2	syl11 34	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
42	41	adantr 485	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
43	42	impcom 412	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
44		ovexd 7446	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
45		eleq1 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
46		oveq1 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)))
47	46	eqeq1d 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
48		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (lastS‘𝑑) = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
49	48	preq2d 4711	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {(lastS‘𝑊), (lastS‘𝑑)} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
50	49	eleq1d 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
51	47, 50	anbi12d 643	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) ↔ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)))
52	45, 51	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ↔ ((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))))
53	48	eqeq2d 2780	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = (lastS‘𝑑) ↔ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
54	52, 53	anbi12d 643	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)) ↔ (((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))))
55	54	bicomd 226	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
56	55	adantl 486	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
57	56	biimpd 232	. . . . . . . . . 10 ⊢ (((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) ∧ 𝑑 = (𝑊 ++ ⟨“𝑟”⟩)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) → ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
58	44, 57	spcimedv 3563	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((((𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)) ∧ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑))))
59	40, 43, 58	mp2and 711	. . . . . . . 8 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)))
60		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
61	60	eqeq1d 2771	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
62		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
63	62	preq2d 4711	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
64	63	eleq1d 2854	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
65	61, 64	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑑 → (((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
66	65	elrab 3659	. . . . . . . . . 10 ⊢ (𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ↔ (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
67	66	anbi1i 635	. . . . . . . . 9 ⊢ ((𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)) ↔ ((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)))
68	67	exbii 1875	. . . . . . . 8 ⊢ (∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)) ↔ ∃𝑑((𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ∧ ((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ 𝑟 = (lastS‘𝑑)))
69	59, 68	sylibr 237	. . . . . . 7 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)))
70		df-rex 3096	. . . . . . 7 ⊢ (∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}𝑟 = (lastS‘𝑑) ↔ ∃𝑑(𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} ∧ 𝑟 = (lastS‘𝑑)))
71	69, 70	sylibr 237	. . . . . 6 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}𝑟 = (lastS‘𝑑))
72	1, 4, 5	wwlksnextwrd 30187	. . . . . . 7 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
73	72	adantr 485	. . . . . 6 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
74	71, 73	rexeqtrrdv 3334	. . . . 5 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (lastS‘𝑑))
75		fveq2 6882	. . . . . . . 8 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
76		fvex 6895	. . . . . . . 8 ⊢ (lastS‘𝑑) ∈ V
77	75, 7, 76	fvmpt 6990	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
78	77	eqeq2d 2780	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝑟 = (𝐹‘𝑑) ↔ 𝑟 = (lastS‘𝑑)))
79	78	rexbiia 3116	. . . . 5 ⊢ (∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑) ↔ ∃𝑑 ∈ 𝐷 𝑟 = (lastS‘𝑑))
80	74, 79	sylibr 237	. . . 4 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
81	12, 80	sylan2b 605	. . 3 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑅) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
82	81	ralrimiva 3163	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
83		dffo3 7098	. 2 ⊢ (𝐹:𝐷–onto→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑)))
84	9, 82, 83	sylanbrc 594	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463  {cpr 4596   ↦ cmpt 5196  ⟶wf 6533  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7411  0cc0 11100  1c1 11101   + caddc 11103  2c2 12295  ℕ₀cn0 12504  ..^cfzo 13682  ♯chash 14366  Word cword 14550  lastSclsw 14599   ++ cconcat 14607  ⟨“cs1 14633   prefix cpfx 14708  Vtxcvtx 29287  Edgcedg 29338   WWalksN cwwlksn 30116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-xnn0 12578  df-z 12592  df-uz 12863  df-rp 13017  df-fz 13536  df-fzo 13683  df-hash 14367  df-word 14551  df-lsw 14600  df-concat 14608  df-s1 14634  df-substr 14679  df-pfx 14709  df-wwlks 30120  df-wwlksn 30121

This theorem is referenced by:  wwlksnextbij0  30191