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Theorem clwwlkf1 27822

Description: Lemma 3 for clwwlkf1o 27824: 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 27820	. 2 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
4	1, 2	clwwlkfv 27821	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥 prefix 𝑁))
5	1, 2	clwwlkfv 27821	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦 prefix 𝑁))
6	4, 5	eqeqan12d 2838	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
7	6	adantl 484	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 prefix 𝑁) = (𝑦 prefix 𝑁)))
8		fveq2 6665	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
9		fveq1 6664	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
10	8, 9	eqeq12d 2837	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑥) = (𝑥‘0)))
11	10, 1	elrab2 3683	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)))
12		fveq2 6665	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (lastS‘𝑤) = (lastS‘𝑦))
13		fveq1 6664	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
14	12, 13	eqeq12d 2837	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((lastS‘𝑤) = (𝑤‘0) ↔ (lastS‘𝑦) = (𝑦‘0)))
15	14, 1	elrab2 3683	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0)))
16	11, 15	anbi12i 628	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑦) = (𝑦‘0))))
17		eqid 2821	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
18		eqid 2821	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
19	17, 18	wwlknp 27615	. . . . . . . . 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 27615	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
21		simprlr 778	. . . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . . . . . 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 11640	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26		ax-1cn 10589	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ ℂ
27		pncan 10886	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
28	27	eqcomd 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
29	25, 26, 28	sylancl 588	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
30		oveq1 7157	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑥) = (𝑁 + 1) → ((♯‘𝑥) − 1) = ((𝑁 + 1) − 1))
31	30	eqcomd 2827	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((♯‘𝑥) − 1))
32	29, 31	sylan9eqr 2878	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((♯‘𝑥) − 1))
33	32	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 prefix 𝑁) = (𝑥 prefix ((♯‘𝑥) − 1)))
34	32	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 prefix 𝑁) = (𝑦 prefix ((♯‘𝑥) − 1)))
35	33, 34	eqeq12d 2837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((♯‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1))))
36	35	ex 415	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) ↔ (𝑥 prefix ((♯‘𝑥) − 1)) = (𝑦 prefix ((♯‘𝑥) − 1)))))
37	36	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word (Vtx‘𝐺))
42		simpll 765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word (Vtx‘𝐺))
43	41, 42	anim12ci 615	. . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . 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 11898	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
46		0nn0 11906	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℕ₀
47	45, 46	jctil 522	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))
48	47	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 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 11639	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
50	49	lep1d 11565	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
51		breq2 5063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
52	50, 51	syl5ibr 248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
53	52	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑥)))
54	53	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . 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 5063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (♯‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
57	50, 56	syl5ibr 248	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
58	57	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (♯‘𝑦)))
59	58	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . . 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 14029	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ₀) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
62	61	ad2ant2rl 747	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑥 prefix 𝑁) = (𝑥 substr ⟨0, 𝑁⟩))
63		pfxval 14029	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑦 ∈ Word (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ₀) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
64	63	ad2ant2l 744	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺)) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑦 prefix 𝑁) = (𝑦 substr ⟨0, 𝑁⟩))
65	62, 64	eqeq12d 2837	. . . . . . . . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . . . . . . . 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 14021	. . . . . . . . . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . 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 13071	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
71	70	biimpri 230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
72	71	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 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))
73		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
74		fveq2 6665	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
75	73, 74	eqeq12d 2837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
76	75	rspcv 3618	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → (𝑥‘0) = (𝑦‘0)))
77	72, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . 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)))
78	69, 77	sylbid 242	. . . . . . . . . . . . . . . . . . . . 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)))
79	78	imp 409	. . . . . . . . . . . . . . . . . . . 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))
80		simpr 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (lastS‘𝑥) = (𝑥‘0))
81		simpr 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑦) = (𝑁 + 1)) ∧ (lastS‘𝑦) = (𝑦‘0)) → (lastS‘𝑦) = (𝑦‘0))
82	80, 81	eqeqan12rd 2840	. . . . . . . . . . . . . . . . . . . . 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)))
83	82	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 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)))
84	79, 83	mpbird 259	. . . . . . . . . . . . . . . . . . 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))
85	24, 40, 84	jca32 518	. . . . . . . . . . . . . . . . . 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))))
86	42	adantl 484	. . . . . . . . . . . . . . . . . . . . . 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))
87	86	adantl 484	. . . . . . . . . . . . . . . . . . . . 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))
88	41	adantr 483	. . . . . . . . . . . . . . . . . . . . . 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))
89	88	adantl 484	. . . . . . . . . . . . . . . . . . . . 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))
90		1red 10636	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
91		nngt0 11662	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
92		0lt1 11156	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 < 1
93	92	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 1)
94	49, 90, 91, 93	addgt0d 11209	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
95		breq2 5063	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 1)))
96	94, 95	syl5ibr 248	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
97	96	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = (𝑁 + 1)) ∧ (lastS‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (♯‘𝑥)))
98	97	adantl 484	. . . . . . . . . . . . . . . . . . . . . 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)))
99	98	impcom 410	. . . . . . . . . . . . . . . . . . . . 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))
100	87, 89, 99	3jca 1124	. . . . . . . . . . . . . . . . . . . 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)))
101	100	adantr 483	. . . . . . . . . . . . . . . . . . 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)))
102		pfxsuff1eqwrdeq 14055	. . . . . . . . . . . . . . . . . . 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)))))
103	101, 102	syl 17	. . . . . . . . . . . . . . . . . 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)))))
104	85, 103	mpbird 259	. . . . . . . . . . . . . . . . 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)
105	104	exp31 422	. . . . . . . . . . . . . . . 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)))
106	105	expdcom 417	. . . . . . . . . . . . . . 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))))
107	106	ex 415	. . . . . . . . . . . . . 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)))))
108	107	3adant3 1128	. . . . . . . . . . . . 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)))))
109	20, 108	syl 17	. . . . . . . . . . . 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)))))
110	109	imp 409	. . . . . . . . . . 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))))
111	110	expdcom 417	. . . . . . . . . 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)))))
112	111	3adant3 1128	. . . . . . . . 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)))))
113	19, 112	syl 17	. . . . . . . 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)))))
114	113	imp31 420	. . . . . . 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)))
115	114	com12 32	. . . . . 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)))
116	16, 115	syl5bi 244	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦)))
117	116	imp 409	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝑥 prefix 𝑁) = (𝑦 prefix 𝑁) → 𝑥 = 𝑦))
118	7, 117	sylbid 242	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
119	118	ralrimivva 3191	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
120		dff13 7007	. 2 ⊢ (𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
121	3, 119, 120	sylanbrc 585	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 {cpr 4563 ⟨cop 4567 class class class wbr 5059 ↦ cmpt 5139 ⟶wf 6346 –1-1→wf1 6347 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 ℕcn 11632 ℕ₀cn0 11891 ..^cfzo 13027 ♯chash 13684 Word cword 13855 lastSclsw 13908 substr csubstr 13996 prefix cpfx 14026 Vtxcvtx 26775 Edgcedg 26826 WWalksN cwwlksn 27598 ClWWalksN cclwwlkn 27796

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 df-s1 13944 df-substr 13997 df-pfx 14027 df-wwlks 27602 df-wwlksn 27603 df-clwwlk 27754 df-clwwlkn 27797

This theorem is referenced by: clwwlkf1o 27824

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