Theorem clwlkclwwlkf1 27160

Description: 𝐹 is a one-to-one function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.)

Hypotheses

Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩))

Assertion

Ref	Expression
clwlkclwwlkf1	⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

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

Proof of Theorem clwlkclwwlkf1

Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkclwwlkf.c	. . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
2		clwlkclwwlkf.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩))
3	1, 2	clwlkclwwlkf 27158	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4	2	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩)))
5		fveq2 6332	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (2nd ‘𝑐) = (2nd ‘𝑥))
6	5	fveq2d 6336	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑥 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑥)))
7	6	oveq1d 6808	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑥)) − 1))
8	7	opeq2d 4546	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩ = ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩)
9	5, 8	oveq12d 6811	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩) = ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩))
10	9	adantl 467	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑐 = 𝑥) → ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩) = ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩))
11		id 22	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
12		ovexd 6825	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) ∈ V)
13	4, 10, 11, 12	fvmptd 6430	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝐹‘𝑥) = ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩))
14	2	a1i 11	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩)))
15		fveq2 6332	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (2nd ‘𝑐) = (2nd ‘𝑦))
16	15	fveq2d 6336	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑦 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑦)))
17	16	oveq1d 6808	. . . . . . . . . 10 ⊢ (𝑐 = 𝑦 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑦)) − 1))
18	17	opeq2d 4546	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩ = ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)
19	15, 18	oveq12d 6811	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩))
20	19	adantl 467	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐶 ∧ 𝑐 = 𝑦) → ((2nd ‘𝑐) substr ⟨0, ((♯‘(2nd ‘𝑐)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩))
21		id 22	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐶)
22		ovexd 6825	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩) ∈ V)
23	14, 20, 21, 22	fvmptd 6430	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝐹‘𝑦) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩))
24	13, 23	eqeqan12d 2787	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)))
25	24	adantl 467	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)))
26		simpl 468	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → 𝑥 ∈ 𝐶)
27	26	adantl 467	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝑥 ∈ 𝐶)
28	27	adantr 466	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → 𝑥 ∈ 𝐶)
29		simplrr 763	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → 𝑦 ∈ 𝐶)
30		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘𝑥) = (1st ‘𝑥)
31		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝑥) = (2nd ‘𝑥)
32	1, 30, 31	clwlkclwwlkflem 27154	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐶 → ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ))
33		wlklenvm1 26752	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → (♯‘(1st ‘𝑥)) = ((♯‘(2nd ‘𝑥)) − 1))
34	33	eqcomd 2777	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
35	34	3ad2ant1 1127	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
36	32, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐶 → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
37	36	adantr 466	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
38	37	opeq2d 4546	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩ = ⟨0, (♯‘(1st ‘𝑥))⟩)
39	38	oveq2d 6809	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩))
40		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘𝑦) = (1st ‘𝑦)
41		eqid 2771	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘𝑦) = (2nd ‘𝑦)
42	1, 40, 41	clwlkclwwlkflem 27154	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝐶 → ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ))
43		wlklenvm1 26752	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → (♯‘(1st ‘𝑦)) = ((♯‘(2nd ‘𝑦)) − 1))
44	43	eqcomd 2777	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
45	44	3ad2ant1 1127	. . . . . . . . . . . . . . 15 ⊢ (((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
46	42, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
47	46	adantl 467	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
48	47	opeq2d 4546	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩ = ⟨0, (♯‘(1st ‘𝑦))⟩)
49	48	oveq2d 6809	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩))
50	39, 49	eqeq12d 2786	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩) ↔ ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩)))
51	50	adantl 467	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩) ↔ ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩)))
52	51	biimpa 462	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩))
53	28, 29, 52	3jca 1122	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩)))
54	1, 30, 31, 40, 41	clwlkclwwlkf1lem2 27155	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩)) → ((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖)))
55		simpl 468	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)))
56	53, 54, 55	3syl 18	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)))
57	1, 30, 31, 40, 41	clwlkclwwlkf1lem3 27156	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2nd ‘𝑥) substr ⟨0, (♯‘(1st ‘𝑥))⟩) = ((2nd ‘𝑦) substr ⟨0, (♯‘(1st ‘𝑦))⟩)) → ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))
58	53, 57	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))
59		simpl 468	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
60		wlkcpr 26759	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Walks‘𝐺) ↔ (1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥))
61	60	biimpri 218	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → 𝑥 ∈ (Walks‘𝐺))
62	61	3ad2ant1 1127	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
63	32, 62	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ (Walks‘𝐺))
64		wlkcpr 26759	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (Walks‘𝐺) ↔ (1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦))
65	64	biimpri 218	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → 𝑦 ∈ (Walks‘𝐺))
66	65	3ad2ant1 1127	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
67	42, 66	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ (Walks‘𝐺))
68	63, 67	anim12i 600	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
69	68	adantl 467	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
70		eqidd 2772	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥)))
71	59, 69, 70	3jca 1122	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
72	71	adantr 466	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
73		uspgr2wlkeq 26777	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))) → (𝑥 = 𝑦 ↔ ((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))))
74	72, 73	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → (𝑥 = 𝑦 ↔ ((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))))
75	56, 58, 74	mpbir2and 692	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩)) → 𝑥 = 𝑦)
76	75	ex 397	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((2nd ‘𝑥) substr ⟨0, ((♯‘(2nd ‘𝑥)) − 1)⟩) = ((2nd ‘𝑦) substr ⟨0, ((♯‘(2nd ‘𝑦)) − 1)⟩) → 𝑥 = 𝑦))
77	25, 76	sylbid 230	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
78	77	ralrimivva 3120	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
79		dff13 6655	. 2 ⊢ (𝐹:𝐶–1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
80	3, 78, 79	sylanbrc 572	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065  Vcvv 3351  ⟨cop 4322   class class class wbr 4786   ↦ cmpt 4863  ⟶wf 6027  –1-1→wf1 6028  ‘cfv 6031  (class class class)co 6793  1^st c1st 7313  2^nd c2nd 7314  0cc0 10138  1c1 10139   ≤ cle 10277   − cmin 10468  ℕcn 11222  ...cfz 12533  ..^cfzo 12673  ♯chash 13321   substr csubstr 13491  USPGraphcuspgr 26265  Walkscwlks 26727  ClWalkscclwlks 26901  ClWWalkscclwwlk 27131

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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-lsw 13496  df-substr 13499  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-wlks 26730  df-clwlks 26902  df-clwwlk 27132

This theorem is referenced by:  clwlkclwwlkf1o  27161