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Theorem clwwlksf1 26917

Description: Lemma 3 for clwwlksbij 26920: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.)

**Hypotheses**
Ref	Expression
clwwlksbij.d	⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
clwwlksbij.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))

**Assertion**
Ref	Expression
clwwlksf1	⊢ (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 clwwlksf1 Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlksbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
2		clwwlksbij.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
3	1, 2	clwwlksf 26915	. 2 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
4	1, 2	clwwlksfv 26916	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
5	1, 2	clwwlksfv 26916	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦 substr ⟨0, 𝑁⟩))
6	4, 5	eqeqan12d 2638	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
7	6	adantl 482	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → ( lastS ‘𝑤) = ( lastS ‘𝑥))
9		fveq1 6190	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
10	8, 9	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑥) = (𝑥‘0)))
11	10, 1	elrab2 3366	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)))
12		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ( lastS ‘𝑤) = ( lastS ‘𝑦))
13		fveq1 6190	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
14	12, 13	eqeq12d 2637	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑦) = (𝑦‘0)))
15	14, 1	elrab2 3366	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)))
16	11, 15	anbi12i 733	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0))))
17		eqid 2622	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
18		eqid 2622	. . . . . . . . . 10 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
19	17, 18	wwlknp 26734	. . . . . . . . 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 26734	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑁 WWalksN 𝐺) → (𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
21		simprlr 803	. . . . . . . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . . . . . 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 2659	. . . . . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (#‘𝑥) = (#‘𝑦))
25		nncn 11028	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
26		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℂ
27		pncan 10287	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
28	27	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
29	25, 26, 28	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
30		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝑥) = (𝑁 + 1) → ((#‘𝑥) − 1) = ((𝑁 + 1) − 1))
31	30	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((#‘𝑥) − 1))
32	29, 31	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((#‘𝑥) − 1))
33	32	opeq2d 4409	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ⟨0, 𝑁⟩ = ⟨0, ((#‘𝑥) − 1)⟩)
34	33	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 substr ⟨0, 𝑁⟩) = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩))
35	33	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))
36	34, 35	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩)))
37	36	ex 450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
38	37	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
39	38	adantl 482	. . . . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
40	39	impcom 446	. . . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩)))
41	40	biimpa 501	. . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))
42		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word (Vtx‘𝐺))
43		simpll 790	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word (Vtx‘𝐺))
44	42, 43	anim12ci 591	. . . . . . . . . . . . . . . . . . . . . . . 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)))
45	44	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 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)))
46		nnnn0 11299	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
47		0nn0 11307	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 0 ∈ ℕ₀
48	46, 47	jctil 560	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))
49	48	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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))
50		nnre 11027	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
51	50	lep1d 10955	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
52		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
53	51, 52	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
54	53	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
55	54	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 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)))
56	55	impcom 446	. . . . . . . . . . . . . . . . . . . . . . 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))
57		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
58	51, 57	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
59	58	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
60	59	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 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)))
61	60	impcom 446	. . . . . . . . . . . . . . . . . . . . . . 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))
62		swrdspsleq 13449	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((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)))
63	45, 49, 56, 61, 62	syl112anc 1330	. . . . . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
64		lbfzo0 12507	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
65	64	biimpri 218	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
66	65	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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))
67		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
68		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
69	67, 68	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
70	69	rspcv 3305	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → (𝑥‘0) = (𝑦‘0)))
71	66, 70	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)))
72	63, 71	sylbid 230	. . . . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → (𝑥‘0) = (𝑦‘0)))
73	72	imp 445	. . . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥‘0) = (𝑦‘0))
74		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ( lastS ‘𝑥) = (𝑥‘0))
75		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ( lastS ‘𝑦) = (𝑦‘0))
76	74, 75	eqeqan12rd 2640	. . . . . . . . . . . . . . . . . . . . 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)))
77	76	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
78	73, 77	mpbird 247	. . . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → ( lastS ‘𝑥) = ( lastS ‘𝑦))
79	24, 41, 78	jca32 558	. . . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦))))
80	43	adantl 482	. . . . . . . . . . . . . . . . . . . . . 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))
81	80	adantl 482	. . . . . . . . . . . . . . . . . . . . 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))
82	42	adantr 481	. . . . . . . . . . . . . . . . . . . . . 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))
83	82	adantl 482	. . . . . . . . . . . . . . . . . . . . 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))
84		1red 10055	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
85		nngt0 11049	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
86		0lt1 10550	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 < 1
87	86	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < 1)
88	50, 84, 85, 87	addgt0d 10602	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
89		breq2 4657	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑥) = (𝑁 + 1) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 1)))
90	88, 89	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
91	90	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
92	91	adantl 482	. . . . . . . . . . . . . . . . . . . . . 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)))
93	92	impcom 446	. . . . . . . . . . . . . . . . . . . . 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))
94	81, 83, 93	3jca 1242	. . . . . . . . . . . . . . . . . . . 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)))
95	94	adantr 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (#‘𝑥)))
96		2swrd1eqwrdeq 13454	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑦 ∈ Word (Vtx‘𝐺) ∧ 0 < (#‘𝑥)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦)))))
97	95, 96	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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦)))))
98	79, 97	mpbird 247	. . . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → 𝑥 = 𝑦)
99	98	exp31 630	. . . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
100	99	expdcom 455	. . . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
101	100	ex 450	. . . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
102	101	3adant3 1081	. . . . . . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
103	20, 102	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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
104	103	imp 445	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
105	104	expdcom 455	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑥) = (𝑁 + 1)) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
106	105	3adant3 1081	. . . . . . . . 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 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
107	19, 106	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑁 WWalksN 𝐺) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
108	107	imp31 448	. . . . . . 7 ⊢ (((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
109	108	com12 32	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (((𝑥 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
110	16, 109	syl5bi 232	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
111	110	imp 445	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))
112	7, 111	sylbid 230	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
113	112	ralrimivva 2971	. 2 ⊢ (𝑁 ∈ ℕ → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
114		dff13 6512	. 2 ⊢ (𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
115	3, 113, 114	sylanbrc 698	1 ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 {cpr 4179 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292 substr csubstr 13295 Vtxcvtx 25874 Edgcedg 25939 WWalksN cwwlksn 26718 ClWWalksN cclwwlksn 26876

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-s1 13302 df-substr 13303 df-wwlks 26722 df-wwlksn 26723 df-clwwlks 26877 df-clwwlksn 26878

This theorem is referenced by: clwwlksf1o 26919

W3C validator