Theorem wwlksnextsurj 30546

Description: Lemma for wwlksnextbij 30548. (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⟶onto𝑅)

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 30488	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
3		simp2 1155	. . 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 30544	. . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
9	2, 3, 8	3syl 19	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷⟶𝑅)
10		preq2 4697	. . . . . 6 ⊢ (𝑛 = 𝑟 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑟})
11	10	eleq1d 2849	. . . . 5 ⊢ (𝑛 = 𝑟 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
12	11, 6	elrab2 3652	. . . 4 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
13	1, 4	wwlksnext 30539	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
14	13	3expb 1138	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
15		s1cl 14790	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ 𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
16		pfxccat1 14893	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
17	15, 16	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
18	17	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
19	18	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
20		oveq2 7459	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 + 1) = (♯‘𝑊) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
21	20	eqcoms 2771	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
22	21	eqeq1d 2765	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
23	22	adantl 487	. . . . . . . . . . . . . . 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 1150	. . . . . . . . . . . . 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 30489	. . . . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
29	28	impcom 413	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊)
30		lswccats1 14824	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → (lastS‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
31	30	eqcomd 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
32	31	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
33	32	3ad2ant3 1153	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
36	35	preq2d 4703	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → {(lastS‘𝑊), 𝑟} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
37	36	eleq1d 2849	. . . . . . . . . . . 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 460	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
40	14, 29, 39	jca32 525	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
43	42	impcom 413	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
44		ovexd 7486	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
45		eleq1 2852	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
46		oveq1 7458	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)))
47	46	eqeq1d 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
48		fveq2 6916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (lastS‘𝑑) = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
49	48	preq2d 4703	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {(lastS‘𝑊), (lastS‘𝑑)} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
50	49	eleq1d 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
51	47, 50	anbi12d 644	. . . . . . . . . . . . . . 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 644	. . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = (lastS‘𝑑) ↔ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
54	52, 53	anbi12d 644	. . . . . . . . . . . . 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 487	. . . . . . . . . . 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 3553	. . . . . . . . 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 712	. . . . . . . 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 7458	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
61	60	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
62		fveq2 6916	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
63	62	preq2d 4703	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
64	63	eleq1d 2849	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
65	61, 64	anbi12d 644	. . . . . . . . . . 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 3648	. . . . . . . . . 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 636	. . . . . . . . 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 1881	. . . . . . . 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 3091	. . . . . . 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 30543	. . . . . . 7 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
73	72	adantr 486	. . . . . 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 3328	. . . . 5 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (lastS‘𝑑))
75		fveq2 6916	. . . . . . . 8 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
76		fvex 6929	. . . . . . . 8 ⊢ (lastS‘𝑑) ∈ V
77	75, 7, 76	fvmpt 7024	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
78	77	eqeq2d 2774	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝑟 = (𝐹‘𝑑) ↔ 𝑟 = (lastS‘𝑑)))
79	78	rexbiia 3111	. . . . 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 606	. . 3 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑅) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
82	81	ralrimiva 3158	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
83		dffo3 7134	. 2 ⊢ (𝐹:𝐷⟶onto𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑)))
84	9, 82, 83	sylanbrc 595	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷⟶onto𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  {cpr 4590   ↦ cmpt 5192  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  (class class class)co 7451  0cc0 11239  1c1 11240   + caddc 11242  2c2 12437  ℕ₀cn0 12646  ..^cfzo 13828  ♯chash 14515  Word cword 14699  lastSclsw 14748   ++ cconcat 14756  ⟨“cs1 14783   prefix cpfx 14862  Vtxcvtx 29631  Edgcedg 29682   WWalksN cwwlksn 30472

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-card 10057  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-n0 12647  df-xnn0 12720  df-z 12734  df-uz 13006  df-rp 13161  df-fz 13680  df-fzo 13829  df-hash 14516  df-word 14700  df-lsw 14749  df-concat 14757  df-s1 14784  df-substr 14831  df-pfx 14863  df-wwlks 30476  df-wwlksn 30477

This theorem is used by:  wwlksnextbij0  30547