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Theorem clwlkclwwlkf1 29807

Description: 𝐹 is a one-to-one function from the nonempty closed walks into the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.)

**Hypotheses**
Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
clwlkclwwlkf.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘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 ≤ (♯‘(1^st ‘𝑤))}
2		clwlkclwwlkf.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
3	1, 2	clwlkclwwlkf 29805	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		fveq2 6891	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (2^nd ‘𝑐) = (2^nd ‘𝑥))
5		2fveq3 6896	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑥)))
6	5	oveq1d 7429	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑥)) − 1))
7	4, 6	oveq12d 7432	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)))
8		id 22	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
9		ovexd 7449	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) ∈ V)
10	2, 7, 8, 9	fvmptd3 7022	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝐹‘𝑥) = ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)))
11		fveq2 6891	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (2^nd ‘𝑐) = (2^nd ‘𝑦))
12		2fveq3 6896	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑦)))
13	12	oveq1d 7429	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑦)) − 1))
14	11, 13	oveq12d 7432	. . . . . . 7 ⊢ (𝑐 = 𝑦 → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)))
15		id 22	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐶)
16		ovexd 7449	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)) ∈ V)
17	2, 14, 15, 16	fvmptd3 7022	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝐹‘𝑦) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)))
18	10, 17	eqeqan12d 2741	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))))
19	18	adantl 481	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))))
20		simplrl 776	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → 𝑥 ∈ 𝐶)
21		simplrr 777	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → 𝑦 ∈ 𝐶)
22		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑥) = (1^st ‘𝑥)
23		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘𝑥) = (2^nd ‘𝑥)
24	1, 22, 23	clwlkclwwlkflem 29801	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐶 → ((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) ∧ ((2^nd ‘𝑥)‘0) = ((2^nd ‘𝑥)‘(♯‘(1^st ‘𝑥))) ∧ (♯‘(1^st ‘𝑥)) ∈ ℕ))
25		wlklenvm1 29423	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) → (♯‘(1^st ‘𝑥)) = ((♯‘(2^nd ‘𝑥)) − 1))
26	25	eqcomd 2733	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) → ((♯‘(2^nd ‘𝑥)) − 1) = (♯‘(1^st ‘𝑥)))
27	26	3ad2ant1 1131	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) ∧ ((2^nd ‘𝑥)‘0) = ((2^nd ‘𝑥)‘(♯‘(1^st ‘𝑥))) ∧ (♯‘(1^st ‘𝑥)) ∈ ℕ) → ((♯‘(2^nd ‘𝑥)) − 1) = (♯‘(1^st ‘𝑥)))
28	24, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐶 → ((♯‘(2^nd ‘𝑥)) − 1) = (♯‘(1^st ‘𝑥)))
29	28	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2^nd ‘𝑥)) − 1) = (♯‘(1^st ‘𝑥)))
30	29	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))))
31		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑦) = (1^st ‘𝑦)
32		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘𝑦) = (2^nd ‘𝑦)
33	1, 31, 32	clwlkclwwlkflem 29801	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → ((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) ∧ ((2^nd ‘𝑦)‘0) = ((2^nd ‘𝑦)‘(♯‘(1^st ‘𝑦))) ∧ (♯‘(1^st ‘𝑦)) ∈ ℕ))
34		wlklenvm1 29423	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) → (♯‘(1^st ‘𝑦)) = ((♯‘(2^nd ‘𝑦)) − 1))
35	34	eqcomd 2733	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) → ((♯‘(2^nd ‘𝑦)) − 1) = (♯‘(1^st ‘𝑦)))
36	35	3ad2ant1 1131	. . . . . . . . . . . . . 14 ⊢ (((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) ∧ ((2^nd ‘𝑦)‘0) = ((2^nd ‘𝑦)‘(♯‘(1^st ‘𝑦))) ∧ (♯‘(1^st ‘𝑦)) ∈ ℕ) → ((♯‘(2^nd ‘𝑦)) − 1) = (♯‘(1^st ‘𝑦)))
37	33, 36	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐶 → ((♯‘(2^nd ‘𝑦)) − 1) = (♯‘(1^st ‘𝑦)))
38	37	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2^nd ‘𝑦)) − 1) = (♯‘(1^st ‘𝑦)))
39	38	oveq2d 7430	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦))))
40	30, 39	eqeq12d 2743	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)) ↔ ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦)))))
41	40	adantl 481	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)) ↔ ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦)))))
42	41	biimpa 476	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦))))
43	20, 21, 42	3jca 1126	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦)))))
44	1, 22, 23, 31, 32	clwlkclwwlkf1lem2 29802	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦)))) → ((♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1^st ‘𝑥)))((2^nd ‘𝑥)‘𝑖) = ((2^nd ‘𝑦)‘𝑖)))
45		simpl 482	. . . . . . 7 ⊢ (((♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1^st ‘𝑥)))((2^nd ‘𝑥)‘𝑖) = ((2^nd ‘𝑦)‘𝑖)) → (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑦)))
46	43, 44, 45	3syl 18	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑦)))
47	1, 22, 23, 31, 32	clwlkclwwlkf1lem3 29803	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦)))) → ∀𝑖 ∈ (0...(♯‘(1^st ‘𝑥)))((2^nd ‘𝑥)‘𝑖) = ((2^nd ‘𝑦)‘𝑖))
48	43, 47	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → ∀𝑖 ∈ (0...(♯‘(1^st ‘𝑥)))((2^nd ‘𝑥)‘𝑖) = ((2^nd ‘𝑦)‘𝑖))
49		simpl 482	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
50		wlkcpr 29430	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥))
51	50	biimpri 227	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) → 𝑥 ∈ (Walks‘𝐺))
52	51	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) ∧ ((2^nd ‘𝑥)‘0) = ((2^nd ‘𝑥)‘(♯‘(1^st ‘𝑥))) ∧ (♯‘(1^st ‘𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
53	24, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ (Walks‘𝐺))
54		wlkcpr 29430	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦))
55	54	biimpri 227	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) → 𝑦 ∈ (Walks‘𝐺))
56	55	3ad2ant1 1131	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) ∧ ((2^nd ‘𝑦)‘0) = ((2^nd ‘𝑦)‘(♯‘(1^st ‘𝑦))) ∧ (♯‘(1^st ‘𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
57	33, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ (Walks‘𝐺))
58	53, 57	anim12i 612	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
59	58	adantl 481	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60		eqidd 2728	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑥)))
61	49, 59, 60	3jca 1126	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑥))))
62	61	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑥))))
63		uspgr2wlkeq 29447	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑥))) → (𝑥 = 𝑦 ↔ ((♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1^st ‘𝑥)))((2^nd ‘𝑥)‘𝑖) = ((2^nd ‘𝑦)‘𝑖))))
64	62, 63	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → (𝑥 = 𝑦 ↔ ((♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1^st ‘𝑥)))((2^nd ‘𝑥)‘𝑖) = ((2^nd ‘𝑦)‘𝑖))))
65	46, 48, 64	mpbir2and 712	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1))) → 𝑥 = 𝑦)
66	65	ex 412	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)) → 𝑥 = 𝑦))
67	19, 66	sylbid 239	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
68	67	ralrimivva 3195	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69		dff13 7259	. 2 ⊢ (𝐹:𝐶–1-1→(ClWWalks‘𝐺) ↔ (𝐹:𝐶⟶(ClWWalks‘𝐺) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
70	3, 68, 69	sylanbrc 582	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 {crab 3427 Vcvv 3469 class class class wbr 5142 ↦ cmpt 5225 ⟶wf 6538 –1-1→wf1 6539 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 0cc0 11130 1c1 11131 ≤ cle 11271 − cmin 11466 ℕcn 12234 ...cfz 13508 ..^cfzo 13651 ♯chash 14313 prefix cpfx 14644 USPGraphcuspgr 28948 Walkscwlks 29397 ClWalkscclwlks 29571 ClWWalkscclwwlk 29778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ifp 1062 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-lsw 14537 df-substr 14615 df-pfx 14645 df-edg 28848 df-uhgr 28858 df-upgr 28882 df-uspgr 28950 df-wlks 29400 df-clwlks 29572 df-clwwlk 29779

This theorem is referenced by: clwlkclwwlkf1o 29808

W3C validator