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Theorem clwwlkf1 30309

Description: Lemma 3 for clwwlkf1o 30311: 𝐹 is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)

**Hypotheses**
Ref	Expression
clwwlkf1o.d	⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
clwwlkf1o.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))

**Assertion**
Ref	Expression
clwwlkf1	⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺))

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑡,𝐷 𝑡,𝐺,𝑤 𝑡,𝑁 Allowed substitution hints: 𝐷(𝑤) 𝐹(𝑤,𝑡)

Proof of Theorem clwwlkf1 Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlkf1o.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}
2		clwwlkf1o.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))
3	1, 2	clwwlkf 30307	. 2 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
4	1, 2	clwwlkfv 30308	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥 prefix 𝑁))
5	1, 2	clwwlkfv 30308	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦 prefix 𝑁))
6	4, 5	eqeqan12d 2779	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
7	6	adantl 486	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
8		fveq2 6871	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
9		fveq1 6870	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
10	8, 9	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑥) = (𝑥‘0)))
11	10, 1	elrab2 3657	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)))
12		fveq2 6871	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (lastS‘𝑤) = (lastS‘𝑦))
13		fveq1 6870	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
14	12, 13	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑦) = (𝑦‘0)))
15	14, 1	elrab2 3657	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)))
16	11, 15	anbi12i 639	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))))
17		eqid 2765	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
18		eqid 2765	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
19	17, 18	wwlknp 30101	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑁 WWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥‘𝑖), (𝑥‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
20	17, 18	wwlknp 30101	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
21		simprlr 791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (♯‘𝑥) = (𝑁 + 1))
22		simpllr 787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (♯‘𝑦) = (𝑁 + 1))
23	21, 22	eqtr4d 2803	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (♯‘𝑥) = (♯‘𝑦))
24	23	ad2antlr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → (♯‘𝑥) = (♯‘𝑦))
25		nncn 12232	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26		ax-1cn 11146	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℂ
27		pncan 11451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
28	27	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
29	25, 26, 28	sylancl 597	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
30		oveq1 7407	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑥) = (𝑁 + 1) → ((♯‘𝑥) − 1) = ((𝑁 + 1) − 1))
31	30	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((♯‘𝑥) − 1))
32	29, 31	sylan9eqr 2822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((♯‘𝑥) − 1))
33	32	oveq2d 7416	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 prefix 𝑁) = (𝑥 prefix ((♯‘𝑥) − 1)))
34	32	oveq2d 7416	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 prefix 𝑁) = (𝑦 prefix ((♯‘𝑥) − 1)))
35	33, 34	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1))))
36	35	ex 417	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
37	36	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
38	37	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
39	38	impcom 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1))))
40	39	biimpa 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))
41		simpll 778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word (Vtx‘𝐺))
42		simpll 778	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word (Vtx‘𝐺))
43	41, 42	anim12ci 625	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)))
44	43	adantl 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)))
45		nnnn0 12502	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
46		0nn0 12510	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℕ₀
47	45, 46	jctil 528	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))
48	47	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))
49		nnre 12231	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50	49	lep1d 12137	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
51		breq2 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
52	50, 51	imbitrrid 249	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
53	52	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
54	53	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
55	54	impcom 412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (♯‘𝑥))
56		breq2 5109	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
57	50, 56	imbitrrid 249	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
58	57	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
59	58	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
60	59	impcom 412	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (♯‘𝑦))
61		pfxval 14701	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ₀) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
62	61	ad2ant2rl 761	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
63		pfxval 14701	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ₀) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
64	63	ad2ant2l 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
65	62, 64	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
66	65	3adant3 1148	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑥) ∧ 𝑁 ≤ (♯‘𝑦))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
67		swrdspsleq 14693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑥) ∧ 𝑁 ≤ (♯‘𝑦))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
68	66, 67	bitrd 282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑥) ∧ 𝑁 ≤ (♯‘𝑦))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
69	44, 48, 55, 60, 68	syl112anc 1397	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
70		lbfzo0 13719	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
71	70	biranri 510	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → 0 ∈ (0..^𝑁))
72		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
73		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
74	72, 73	eqeq12d 2781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
75	74	rspcv 3580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → (𝑥‘0) = (𝑦‘0)))
76	71, 75	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → (∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → (𝑥‘0) = (𝑦‘0)))
77	69, 76	sylbid 243	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → (𝑥‘0) = (𝑦‘0)))
78	77	imp 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥‘0) = (𝑦‘0))
79		simpr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (lastS‘𝑥) = (𝑥‘0))
80		simpr 489	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (lastS‘𝑦) = (𝑦‘0))
81	79, 80	eqeqan12rd 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → ((lastS‘𝑥) = (lastS‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
82	81	ad2antlr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → ((lastS‘𝑥) = (lastS‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
83	78, 82	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → (lastS‘𝑥) = (lastS‘𝑦))
84	24, 40, 83	jca32 524	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → ((♯‘𝑥) = (♯‘𝑦) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)) ∧ (lastS‘𝑥) = (lastS‘𝑦))))
85	42	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → 𝑥 ∈ Word (Vtx‘𝐺))
86	85	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → 𝑥 ∈ Word (Vtx‘𝐺))
87	41	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → 𝑦 ∈ Word (Vtx‘𝐺))
88	87	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → 𝑦 ∈ Word (Vtx‘𝐺))
89		1red 11197	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
90		nngt0 12258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
91		0lt1 11724	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 < 1
92	91	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 1)
93	49, 89, 90, 92	addgt0d 11777	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
94		breq2 5109	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 1)))
95	93, 94	imbitrrid 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
96	95	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
97	96	adantl 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
98	97	impcom 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → 0 < (♯‘𝑥))
99	86, 88, 98	3jca 1144	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (♯‘𝑥)))
100	99	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (♯‘𝑥)))
101		pfxsuff1eqwrdeq 14726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (♯‘𝑥)) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)) ∧ (lastS‘𝑥) = (lastS‘𝑦)))))
102	100, 101	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → (𝑥 = 𝑦 ↔ ((♯‘𝑥) = (♯‘𝑦) ∧ ((𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)) ∧ (lastS‘𝑥) = (lastS‘𝑦)))))
103	84, 102	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)))) ∧ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)) → 𝑥 = 𝑦)
104	103	exp31 424	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → ((((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
105	104	expdcom 419	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))))
106	105	ex 417	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) → ((lastS‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
107	106	3adant3 1148	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((lastS‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
108	20, 107	syl 18	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → ((lastS‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
109	108	imp 411	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))))
110	109	expdcom 419	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) → ((lastS‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
111	110	3adant3 1148	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥‘𝑖), (𝑥‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ((lastS‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
112	19, 111	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑁 WWalksN 𝐺) → ((lastS‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))))
113	112	imp31 422	. . . . . . 7 ⊢ (((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
114	113	com12 33	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
115	16, 114	biimtrid 245	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
116	115	imp 411	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))
117	7, 116	sylbid 243	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
118	117	ralrimivva 3208	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
119		dff13 7242	. 2 ⊢ (𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
120	3, 118, 119	sylanbrc 594	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1563 ∈ wcel 2145 ∀wral 3079 {crab 3417 {cpr 4587 ⟨cop 4591 class class class wbr 5105 ↦ cmpt 5186 ⟶wf 6521 –1-1→wf1 6522 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 0cc0 11088 1c1 11089 + caddc 11091 < clt 11231 ≤ cle 11232 − cmin 11429 ℕcn 12224 ℕ₀cn0 12495 ..^cfzo 13673 ♯chash 14357 Word cword 14540 lastSclsw 14589 substr csubstr 14668 prefix cpfx 14698 Vtxcvtx 29255 Edgcedg 29306 WWalksN cwwlksn 30084 ClWWalksN cclwwlkn 30284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-n0 12496 df-xnn0 12569 df-z 12583 df-uz 12854 df-fz 13527 df-fzo 13674 df-hash 14358 df-word 14541 df-lsw 14590 df-s1 14624 df-substr 14669 df-pfx 14699 df-wwlks 30088 df-wwlksn 30089 df-clwwlk 30242 df-clwwlkn 30285

This theorem is referenced by: clwwlkf1o 30311

W3C validator