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Theorem clwlkclwwlkf1 29263

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 29261	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		fveq2 6892	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (2^nd ‘𝑐) = (2^nd ‘𝑥))
5		2fveq3 6897	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑥)))
6	5	oveq1d 7424	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑥)) − 1))
7	4, 6	oveq12d 7427	. . . . . . 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 7444	. . . . . . 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 6892	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (2^nd ‘𝑐) = (2^nd ‘𝑦))
12		2fveq3 6897	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑦)))
13	12	oveq1d 7424	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑦)) − 1))
14	11, 13	oveq12d 7427	. . . . . . 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 7444	. . . . . . 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 2747	. . . . 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 483	. . . 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 2733	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑥) = (1^st ‘𝑥)
23		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘𝑥) = (2^nd ‘𝑥)
24	1, 22, 23	clwlkclwwlkflem 29257	. . . . . . . . . . . . . 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 28879	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) → (♯‘(1^st ‘𝑥)) = ((♯‘(2^nd ‘𝑥)) − 1))
26	25	eqcomd 2739	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑥)(Walks‘𝐺)(2^nd ‘𝑥) → ((♯‘(2^nd ‘𝑥)) − 1) = (♯‘(1^st ‘𝑥)))
27	26	3ad2ant1 1134	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2^nd ‘𝑥)) − 1) = (♯‘(1^st ‘𝑥)))
30	29	oveq2d 7425	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2^nd ‘𝑥) prefix ((♯‘(2^nd ‘𝑥)) − 1)) = ((2^nd ‘𝑥) prefix (♯‘(1^st ‘𝑥))))
31		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑦) = (1^st ‘𝑦)
32		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ (2^nd ‘𝑦) = (2^nd ‘𝑦)
33	1, 31, 32	clwlkclwwlkflem 29257	. . . . . . . . . . . . . 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 28879	. . . . . . . . . . . . . . . 16 ⊢ ((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) → (♯‘(1^st ‘𝑦)) = ((♯‘(2^nd ‘𝑦)) − 1))
35	34	eqcomd 2739	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑦)(Walks‘𝐺)(2^nd ‘𝑦) → ((♯‘(2^nd ‘𝑦)) − 1) = (♯‘(1^st ‘𝑦)))
36	35	3ad2ant1 1134	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2^nd ‘𝑦)) − 1) = (♯‘(1^st ‘𝑦)))
39	38	oveq2d 7425	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2^nd ‘𝑦) prefix ((♯‘(2^nd ‘𝑦)) − 1)) = ((2^nd ‘𝑦) prefix (♯‘(1^st ‘𝑦))))
40	30, 39	eqeq12d 2749	. . . . . . . . . 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 483	. . . . . . . . 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 478	. . . . . . . 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 1129	. . . . . . 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 29258	. . . . . . 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 484	. . . . . . 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 29259	. . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
50		wlkcpr 28886	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 28886	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 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 614	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
59	58	adantl 483	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60		eqidd 2734	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑥)))
61	49, 59, 60	3jca 1129	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1^st ‘𝑥)) = (♯‘(1^st ‘𝑥))))
62	61	adantr 482	. . . . . . 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 28903	. . . . . . 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 414	. . . 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 3201	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69		dff13 7254	. 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 584	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 class class class wbr 5149 ↦ cmpt 5232 ⟶wf 6540 –1-1→wf1 6541 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 0cc0 11110 1c1 11111 ≤ cle 11249 − cmin 11444 ℕcn 12212 ...cfz 13484 ..^cfzo 13627 ♯chash 14290 prefix cpfx 14620 USPGraphcuspgr 28408 Walkscwlks 28853 ClWalkscclwlks 29027 ClWWalkscclwwlk 29234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-lsw 14513 df-substr 14591 df-pfx 14621 df-edg 28308 df-uhgr 28318 df-upgr 28342 df-uspgr 28410 df-wlks 28856 df-clwlks 29028 df-clwwlk 29235

This theorem is referenced by: clwlkclwwlkf1o 29264

W3C validator