Theorem wwlksnextsurj 29837

Description: Lemma for wwlksnextbij 29839. (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 29779	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
3		simp2 1137	. . 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 29835	. . 3 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
9	2, 3, 8	3syl 18	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷⟶𝑅)
10		preq2 4701	. . . . . 6 ⊢ (𝑛 = 𝑟 → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), 𝑟})
11	10	eleq1d 2814	. . . . 5 ⊢ (𝑛 = 𝑟 → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
12	11, 6	elrab2 3665	. . . 4 ⊢ (𝑟 ∈ 𝑅 ↔ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸))
13	1, 4	wwlksnext 29830	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
14	13	3expb 1120	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
15		s1cl 14574	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ 𝑉 → ⟨“𝑟”⟩ ∈ Word 𝑉)
16		pfxccat1 14674	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑟”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
17	15, 16	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊)
18	17	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
19	18	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
20		oveq2 7398	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 + 1) = (♯‘𝑊) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
21	20	eqcoms 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) = (𝑁 + 1) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)))
22	21	eqeq1d 2732	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) = (𝑁 + 1) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
23	22	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (♯‘𝑊)) = 𝑊))
24	19, 23	sylibrd 259	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1)) → (𝑟 ∈ 𝑉 → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
25	24	3adant3 1132	. . . . . . . . . . . . 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 29780	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
27	25, 26	syl11 33	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
28	27	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
29	28	impcom 407	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊)
30		lswccats1 14606	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → (lastS‘(𝑊 ++ ⟨“𝑟”⟩)) = 𝑟)
31	30	eqcomd 2736	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑟 ∈ 𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
32	31	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑉 → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
33	32	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
34	2, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑟 ∈ 𝑉 → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
35	34	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
36	35	preq2d 4707	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → {(lastS‘𝑊), 𝑟} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
37	36	eleq1d 2814	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
38	37	biimpd 229	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑉) → ({(lastS‘𝑊), 𝑟} ∈ 𝐸 → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
39	38	impr 454	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸)
40	14, 29, 39	jca32 515	. . . . . . . . 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 33	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝑉 → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
43	42	impcom 407	. . . . . . . . 9 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
44		ovexd 7425	. . . . . . . . . 10 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → (𝑊 ++ ⟨“𝑟”⟩) ∈ V)
45		eleq1 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ (𝑊 ++ ⟨“𝑟”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺)))
46		oveq1 7397	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑑 prefix (𝑁 + 1)) = ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)))
47	46	eqeq1d 2732	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ((𝑑 prefix (𝑁 + 1)) = 𝑊 ↔ ((𝑊 ++ ⟨“𝑟”⟩) prefix (𝑁 + 1)) = 𝑊))
48		fveq2 6861	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (lastS‘𝑑) = (lastS‘(𝑊 ++ ⟨“𝑟”⟩)))
49	48	preq2d 4707	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → {(lastS‘𝑊), (lastS‘𝑑)} = {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))})
50	49	eleq1d 2814	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → ({(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘(𝑊 ++ ⟨“𝑟”⟩))} ∈ 𝐸))
51	47, 50	anbi12d 632	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑊 ++ ⟨“𝑟”⟩) → (𝑟 = (lastS‘𝑑) ↔ 𝑟 = (lastS‘(𝑊 ++ ⟨“𝑟”⟩))))
54	52, 53	anbi12d 632	. . . . . . . . . . . . 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 223	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 229	. . . . . . . . . 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 3564	. . . . . . . . 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 699	. . . . . . . 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 7397	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
61	60	eqeq1d 2732	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
62		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
63	62	preq2d 4707	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
64	63	eleq1d 2814	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
65	61, 64	anbi12d 632	. . . . . . . . . . 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 3662	. . . . . . . . . 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 624	. . . . . . . . 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 1848	. . . . . . . 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 234	. . . . . . 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 3055	. . . . . . 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 234	. . . . . 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 29834	. . . . . . 7 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
73	72	adantr 480	. . . . . 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 3306	. . . . 5 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (lastS‘𝑑))
75		fveq2 6861	. . . . . . . 8 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
76		fvex 6874	. . . . . . . 8 ⊢ (lastS‘𝑑) ∈ V
77	75, 7, 76	fvmpt 6971	. . . . . . 7 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
78	77	eqeq2d 2741	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝑟 = (𝐹‘𝑑) ↔ 𝑟 = (lastS‘𝑑)))
79	78	rexbiia 3075	. . . . 5 ⊢ (∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑) ↔ ∃𝑑 ∈ 𝐷 𝑟 = (lastS‘𝑑))
80	74, 79	sylibr 234	. . . 4 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑟 ∈ 𝑉 ∧ {(lastS‘𝑊), 𝑟} ∈ 𝐸)) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
81	12, 80	sylan2b 594	. . 3 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑟 ∈ 𝑅) → ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
82	81	ralrimiva 3126	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑))
83		dffo3 7077	. 2 ⊢ (𝐹:𝐷–onto→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑟 ∈ 𝑅 ∃𝑑 ∈ 𝐷 𝑟 = (𝐹‘𝑑)))
84	9, 82, 83	sylanbrc 583	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450  {cpr 4594   ↦ cmpt 5191  ⟶wf 6510  –onto→wfo 6512  ‘cfv 6514  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078  2c2 12248  ℕ₀cn0 12449  ..^cfzo 13622  ♯chash 14302  Word cword 14485  lastSclsw 14534   ++ cconcat 14542  ⟨“cs1 14567   prefix cpfx 14642  Vtxcvtx 28930  Edgcedg 28981   WWalksN cwwlksn 29763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-lsw 14535  df-concat 14543  df-s1 14568  df-substr 14613  df-pfx 14643  df-wwlks 29767  df-wwlksn 29768

This theorem is referenced by:  wwlksnextbij0  29838