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Theorem clwwlkf1 26324

Description: Lemma 3 for clwwlkbij 26327: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)

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

**Assertion**
Ref	Expression
clwwlkf1	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → 𝐹:𝐷–1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Distinct variable groups: 𝑤,𝐸 𝑤,𝑁 𝑤,𝑉 𝑡,𝐷 𝑡,𝐸,𝑤 𝑡,𝑁 𝑡,𝑉 𝑡,𝑋 𝑡,𝑌 Allowed substitution hints: 𝐷(𝑤) 𝐹(𝑤,𝑡) 𝑋(𝑤) 𝑌(𝑤)

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

Step	Hyp	Ref	Expression
1		clwwlkbij.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}
2		clwwlkbij.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))
3	1, 2	clwwlkf 26322	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → 𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁))
4	1, 2	clwwlkfv 26323	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (𝑥 substr ⟨0, 𝑁⟩))
5	1, 2	clwwlkfv 26323	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 → (𝐹‘𝑦) = (𝑦 substr ⟨0, 𝑁⟩))
6	4, 5	eqeqan12d 2626	. . . . 5 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
7	6	adantl 481	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)))
8		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → ( lastS ‘𝑤) = ( lastS ‘𝑥))
9		fveq1 6102	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤‘0) = (𝑥‘0))
10	8, 9	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑥) = (𝑥‘0)))
11	10, 1	elrab2 3333	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)))
12		fveq2 6103	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ( lastS ‘𝑤) = ( lastS ‘𝑦))
13		fveq1 6102	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑤‘0) = (𝑦‘0))
14	12, 13	eqeq12d 2625	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (( lastS ‘𝑤) = (𝑤‘0) ↔ ( lastS ‘𝑦) = (𝑦‘0)))
15	14, 1	elrab2 3333	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)))
16	11, 15	anbi12i 729	. . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) ↔ ((𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0))))
17		wwlknimp 26215	. . . . . . . . 9 ⊢ (𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥‘𝑖), (𝑥‘(𝑖 + 1))} ∈ ran 𝐸))
18		wwlknimp 26215	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸))
19		simprlr 799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑥) = (𝑁 + 1))
20		simpllr 795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑦) = (𝑁 + 1))
21	19, 20	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (#‘𝑥) = (#‘𝑦))
22	21	ad2antlr 759	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (#‘𝑥) = (#‘𝑦))
23		nncn 10905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
24		ax-1cn 9873	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 1 ∈ ℂ
25		pncan 10166	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
26	25	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → 𝑁 = ((𝑁 + 1) − 1))
27	23, 24, 26	sylancl 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ → 𝑁 = ((𝑁 + 1) − 1))
28		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((#‘𝑥) = (𝑁 + 1) → ((#‘𝑥) − 1) = ((𝑁 + 1) − 1))
29	28	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝑥) = (𝑁 + 1) → ((𝑁 + 1) − 1) = ((#‘𝑥) − 1))
30	27, 29	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → 𝑁 = ((#‘𝑥) − 1))
31	30	opeq2d 4347	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ⟨0, 𝑁⟩ = ⟨0, ((#‘𝑥) − 1)⟩)
32	31	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑥 substr ⟨0, 𝑁⟩) = (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩))
33	31	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → (𝑦 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))
34	32, 33	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((#‘𝑥) = (𝑁 + 1) ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩)))
35	34	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ (𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩))))
36	35	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩))))
37	36	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩))))
38	37	impcom 445	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩)))
39	38	biimpa 500	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)⟩))
40		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → 𝑦 ∈ Word 𝑉)
41		simpll 786	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → 𝑥 ∈ Word 𝑉)
42	40, 41	anim12ci 589	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉))
43	42	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉))
44		nnnn0 11176	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
45		0nn0 11184	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ∈ ℕ₀
46	44, 45	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))
47	46	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))
48		nnre 10904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
49	48	lep1d 10834	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ → 𝑁 ≤ (𝑁 + 1))
50		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑥) ↔ 𝑁 ≤ (𝑁 + 1)))
51	49, 50	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
52	51	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
53	52	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑥)))
54	53	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (#‘𝑥))
55		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑦) = (𝑁 + 1) → (𝑁 ≤ (#‘𝑦) ↔ 𝑁 ≤ (𝑁 + 1)))
56	49, 55	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑦) = (𝑁 + 1) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
57	56	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
58	57	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 𝑁 ≤ (#‘𝑦)))
59	58	impcom 445	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑁 ≤ (#‘𝑦))
60		swrdspsleq 13301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉) ∧ (0 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (#‘𝑥) ∧ 𝑁 ≤ (#‘𝑦))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
61	43, 47, 54, 59, 60	syl112anc 1322	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) ↔ ∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
62		lbfzo0 12375	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0 ∈ (0..^𝑁) ↔ 𝑁 ∈ ℕ)
63	62	biimpri 217	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → 0 ∈ (0..^𝑁))
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 0 ∈ (0..^𝑁))
65		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 0 → (𝑥‘𝑖) = (𝑥‘0))
66		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 = 0 → (𝑦‘𝑖) = (𝑦‘0))
67	65, 66	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 0 → ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑥‘0) = (𝑦‘0)))
68	67	rspcv 3278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 ∈ (0..^𝑁) → (∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → (𝑥‘0) = (𝑦‘0)))
69	64, 68	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (∀𝑖 ∈ (0..^𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → (𝑥‘0) = (𝑦‘0)))
70	61, 69	sylbid 229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → (𝑥‘0) = (𝑦‘0)))
71	70	imp 444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥‘0) = (𝑦‘0))
72		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ( lastS ‘𝑥) = (𝑥‘0))
73		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ( lastS ‘𝑦) = (𝑦‘0))
74	72, 73	eqeqan12rd 2628	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
75	74	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (( lastS ‘𝑥) = ( lastS ‘𝑦) ↔ (𝑥‘0) = (𝑦‘0)))
76	71, 75	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → ( lastS ‘𝑥) = ( lastS ‘𝑦))
77	22, 39, 76	jca32 556	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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))))
78	41	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → 𝑥 ∈ Word 𝑉)
79	78	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑥 ∈ Word 𝑉)
80	40	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → 𝑦 ∈ Word 𝑉)
81	80	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 𝑦 ∈ Word 𝑉)
82		1red 9934	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℝ)
83		nngt0 10926	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
84		0lt1 10429	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 < 1
85	84	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ → 0 < 1)
86	48, 82, 83, 85	addgt0d 10481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 0 < (𝑁 + 1))
87		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑥) = (𝑁 + 1) → (0 < (#‘𝑥) ↔ 0 < (𝑁 + 1)))
88	86, 87	syl5ibr 235	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑥) = (𝑁 + 1) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
89	88	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
90	89	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → (𝑁 ∈ ℕ → 0 < (#‘𝑥)))
91	90	impcom 445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → 0 < (#‘𝑥))
92	79, 81, 91	3jca 1235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) → (𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)))
93	92	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → (𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)))
94		2swrd1eqwrdeq 13306	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 𝑦 ∈ Word 𝑉 ∧ 0 < (#‘𝑥)) → (𝑥 = 𝑦 ↔ ((#‘𝑥) = (#‘𝑦) ∧ ((𝑥 substr ⟨0, ((#‘𝑥) − 1)⟩) = (𝑦 substr ⟨0, ((#‘𝑥) − 1)⟩) ∧ ( lastS ‘𝑥) = ( lastS ‘𝑦)))))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘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)))))
96	77, 95	mpbird 246	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)))) ∧ (𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩)) → 𝑥 = 𝑦)
97	96	exp31 628	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
98	97	3ad2ant3 1077	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) ∧ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
99	98	expdcom 454	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
100	99	ex 449	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1)) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
101	100	3adant3 1074	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑦‘𝑖), (𝑦‘(𝑖 + 1))} ∈ ran 𝐸) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
102	18, 101	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (( lastS ‘𝑦) = (𝑦‘0) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
103	102	imp 444	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → (((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) ∧ ( lastS ‘𝑥) = (𝑥‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))))
104	103	expdcom 454	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1)) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
105	104	3adant3 1074	. . . . . . . . 9 ⊢ ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑥‘𝑖), (𝑥‘(𝑖 + 1))} ∈ ran 𝐸) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
106	17, 105	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (( lastS ‘𝑥) = (𝑥‘0) → ((𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0)) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))))
107	106	imp31 447	. . . . . . 7 ⊢ (((𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
108	107	com12 32	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → (((𝑥 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑥) = (𝑥‘0)) ∧ (𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ( lastS ‘𝑦) = (𝑦‘0))) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
109	16, 108	syl5bi 231	. . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦)))
110	109	imp 444	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝑥 substr ⟨0, 𝑁⟩) = (𝑦 substr ⟨0, 𝑁⟩) → 𝑥 = 𝑦))
111	7, 110	sylbid 229	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
112	111	ralrimivva 2954	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
113		dff13 6416	. 2 ⊢ (𝐹:𝐷–1-1→((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐷 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
114	3, 112, 113	sylanbrc 695	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑁 ∈ ℕ) → 𝐹:𝐷–1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 {cpr 4127 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 lastS clsw 13147 substr csubstr 13150 WWalksN cwwlkn 26206 ClWWalksN cclwwlkn 26277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-s1 13157 df-substr 13158 df-wwlk 26207 df-wwlkn 26208 df-clwwlk 26279 df-clwwlkn 26280

This theorem is referenced by: clwwlkf1o 26326

W3C validator