Theorem clwlksfoclwwlkOLD 27244

Description: Obsolete version of clwlkclwwlkfo 27159 as of 25-May-2022. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
clwlksfclwwlk.1	⊢ 𝐴 = (1st ‘𝑐)
clwlksfclwwlk.2	⊢ 𝐵 = (2nd ‘𝑐)
clwlksfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))

Assertion

Ref	Expression
clwlksfoclwwlkOLD	⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–onto→(𝑁 ClWWalksN 𝐺))

Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝐶,𝑐   𝐹,𝑐

Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksfoclwwlkOLD

Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . 3 ⊢ 𝐴 = (1st ‘𝑐)
2		clwlksfclwwlk.2	. . 3 ⊢ 𝐵 = (2nd ‘𝑐)
3		clwlksfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4		clwlksfclwwlk.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
5	1, 2, 3, 4	clwlksfclwwlkOLD 27243	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))
6		eqid 2771	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7	6	clwwlknbp 27190	. . . . 5 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))
8	7	adantl 467	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))
9		isclwwlkn 27180	. . . . . . 7 ⊢ (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑤 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑤) = 𝑁))
10		fusgrusgr 26437	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
11		usgruspgr 26295	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
12	10, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph)
13	12	adantr 466	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USPGraph)
14	13	adantr 466	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 𝐺 ∈ USPGraph)
15		simprl 754	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 𝑤 ∈ Word (Vtx‘𝐺))
16		eleq1 2838	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑤) = 𝑁 → ((♯‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
17		prmnn 15595	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑤) ∈ ℙ → (♯‘𝑤) ∈ ℕ)
18	17	nnge1d 11269	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑤) ∈ ℙ → 1 ≤ (♯‘𝑤))
19	16, 18	syl6bir 244	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (♯‘𝑤)))
20	19	adantl 467	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → (𝑁 ∈ ℙ → 1 ≤ (♯‘𝑤)))
21	20	com12 32	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℙ → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → 1 ≤ (♯‘𝑤)))
22	21	adantl 467	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → 1 ≤ (♯‘𝑤)))
23	22	imp 393	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 1 ≤ (♯‘𝑤))
24		eqid 2771	. . . . . . . . . . 11 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	6, 24	clwlkclwwlk2 27153	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
26	14, 15, 23, 25	syl3anc 1476	. . . . . . . . 9 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
27	26	bicomd 213	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ ∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
28	27	anbi1d 615	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → ((𝑤 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑤) = 𝑁) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (♯‘𝑤) = 𝑁)))
29	9, 28	syl5bb 272	. . . . . 6 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (♯‘𝑤) = 𝑁)))
30		df-br 4788	. . . . . . . . 9 ⊢ (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
31		simpl 468	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
32		prmnn 15595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
33	32	nnge1d 11269	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℙ → 1 ≤ 𝑁)
34	33	ad2antlr 706	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 1 ≤ 𝑁)
35		breq2 4791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑤) = 𝑁 → (1 ≤ (♯‘𝑤) ↔ 1 ≤ 𝑁))
36	35	ad2antll 708	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (1 ≤ (♯‘𝑤) ↔ 1 ≤ 𝑁))
37	34, 36	mpbird 247	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 1 ≤ (♯‘𝑤))
38	15, 37	jca 501	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
39		clwlkwlk 26906	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺))
40		wlklenvclwlk 26786	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝑓) = (♯‘𝑤)))
41	38, 39, 40	syl2im 40	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤)))
42	41	impcom 394	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (♯‘𝑓) = (♯‘𝑤))
43		vex 3354	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑓 ∈ V
44		ovex 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
45	43, 44	op1st 7327	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓
46	45	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
47	46	fveq2d 6337	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓))
48	47	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓))
49		eqcom 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑤) = 𝑁 ↔ 𝑁 = (♯‘𝑤))
50	49	biimpi 206	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑤) = 𝑁 → 𝑁 = (♯‘𝑤))
51	50	ad2antll 708	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 𝑁 = (♯‘𝑤))
52	51	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → 𝑁 = (♯‘𝑤))
53	42, 48, 52	3eqtr4d 2815	. . . . . . . . . . . . 13 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
54	1	fveq2i 6336	. . . . . . . . . . . . . . . 16 ⊢ (♯‘𝐴) = (♯‘(1st ‘𝑐))
55	54	eqeq1i 2776	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐴) = 𝑁 ↔ (♯‘(1st ‘𝑐)) = 𝑁)
56		fveq2 6333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
57	56	fveq2d 6337	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(1st ‘𝑐)) = (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
58	57	eqeq1d 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(1st ‘𝑐)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
59	55, 58	syl5bb 272	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘𝐴) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
60	59, 3	elrab2 3518	. . . . . . . . . . . . 13 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
61	31, 53, 60	sylanbrc 572	. . . . . . . . . . . 12 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
62	42	adantr 466	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = (♯‘𝑤))
63	62	opeq2d 4547	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨0, (♯‘𝑓)⟩ = ⟨0, (♯‘𝑤)⟩)
64	63	oveq2d 6812	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
65		simpr 471	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
66	41	adantl 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤)))
67		eqeq2 2782	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 = (♯‘𝑤) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑓) = (♯‘𝑤)))
68	67	eqcoms 2779	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑤) = 𝑁 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑓) = (♯‘𝑤)))
69	68	imbi2d 329	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((♯‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤))))
70	69	ad2antll 708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤))))
71	70	adantl 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤))))
72	66, 71	mpbird 247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁))
73	72	imp 393	. . . . . . . . . . . . . . . . . 18 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = 𝑁)
74	45	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
75	74	fveq2d 6337	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓))
76	57, 75	eqtrd 2805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(1st ‘𝑐)) = (♯‘𝑓))
77	76	eqeq1d 2773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(1st ‘𝑐)) = 𝑁 ↔ (♯‘𝑓) = 𝑁))
78	55, 77	syl5bb 272	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘𝐴) = 𝑁 ↔ (♯‘𝑓) = 𝑁))
79	78, 3	elrab2 3518	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (♯‘𝑓) = 𝑁))
80	65, 73, 79	sylanbrc 572	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
81		ovex 6827	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩) ∈ V
82	54	opeq2i 4544	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⟨0, (♯‘𝐴)⟩ = ⟨0, (♯‘(1st ‘𝑐))⟩
83	2, 82	oveq12i 6808	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = ((2nd ‘𝑐) substr ⟨0, (♯‘(1st ‘𝑐))⟩)
84		fveq2 6333	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd ‘𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
85	57	opeq2d 4547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (♯‘(1st ‘𝑐))⟩ = ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
86	84, 85	oveq12d 6814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) substr ⟨0, (♯‘(1st ‘𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
87	43, 44	op2nd 7328	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
88	45	fveq2i 6336	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓)
89	88	opeq2i 4544	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (♯‘𝑓)⟩
90	87, 89	oveq12i 6808	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩)
91	86, 90	syl6eq 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd ‘𝑐) substr ⟨0, (♯‘(1st ‘𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
92	83, 91	syl5eq 2817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (♯‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
93	92, 4	fvmptg 6424	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
94	80, 81, 93	sylancl 574	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
95	38	ad2antlr 706	. . . . . . . . . . . . . . . . 17 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
96		simpl 468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → 𝑤 ∈ Word (Vtx‘𝐺))
97		wrdsymb1 13539	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
98	97	s1cld 13583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
99		eqidd 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (♯‘𝑤) = (♯‘𝑤))
100		swrdccatid 13706	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩) = 𝑤)
101	96, 98, 99, 100	syl3anc 1476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩) = 𝑤)
102	101	eqcomd 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
103	95, 102	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
104	64, 94, 103	3eqtr4rd 2816	. . . . . . . . . . . . . . 15 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
105	104	ex 397	. . . . . . . . . . . . . 14 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
106	105	adantr 466	. . . . . . . . . . . . 13 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
107		fveq2 6333	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹‘𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
108	107	eqeq2d 2781	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹‘𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
109	108	imbi2d 329	. . . . . . . . . . . . . 14 ⊢ (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
110	109	adantl 467	. . . . . . . . . . . . 13 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
111	106, 110	mpbird 247	. . . . . . . . . . . 12 ⊢ (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘𝑐)))
112	61, 111	rspcimedv 3462	. . . . . . . . . . 11 ⊢ ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
113	112	ex 397	. . . . . . . . . 10 ⊢ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))))
114	113	pm2.43b 55	. . . . . . . . 9 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
115	30, 114	syl5bi 232	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
116	115	exlimdv 2013	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
117	116	adantrd 479	. . . . . 6 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → ((∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (♯‘𝑤) = 𝑁) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
118	29, 117	sylbid 230	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
119	118	impancom 439	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
120	8, 119	mpd 15	. . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
121	120	ralrimiva 3115	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐))
122		dffo3 6519	. 2 ⊢ (𝐹:𝐶–onto→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐 ∈ 𝐶 𝑤 = (𝐹‘𝑐)))
123	5, 121, 122	sylanbrc 572	1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶–onto→(𝑁 ClWWalksN 𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351  ⟨cop 4323   class class class wbr 4787   ↦ cmpt 4864  ⟶wf 6026  –onto→wfo 6028  ‘cfv 6030  (class class class)co 6796  1^st c1st 7317  2^nd c2nd 7318  0cc0 10142  1c1 10143   ≤ cle 10281  ♯chash 13321  Word cword 13487   ++ cconcat 13489  ⟨“cs1 13490   substr csubstr 13491  ℙcprime 15592  Vtxcvtx 26095  iEdgciedg 26096  USPGraphcuspgr 26265  USGraphcusgr 26266  FinUSGraphcfusgr 26431  Walkscwlks 26727  ClWalkscclwlks 26901  ClWWalkscclwwlk 27131   ClWWalksN cclwwlkn 27174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-prm 15593  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-usgr 26268  df-fusgr 26432  df-wlks 26730  df-clwlks 26902  df-clwwlk 27132  df-clwwlkn 27176

This theorem is referenced by:  clwlksf1oclwwlkOLD  27251