Theorem clwlkclwwlkf1 30219

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 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(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) prefix ((♯‘(2nd ‘𝑐)) − 1)))
3	1, 2	clwlkclwwlkf 30217	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		fveq2 6867	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (2nd ‘𝑐) = (2nd ‘𝑥))
5		2fveq3 6872	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑥)))
6	5	oveq1d 7411	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑥)) − 1))
7	4, 6	oveq12d 7414	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)))
8		id 22	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
9		ovexd 7431	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) ∈ V)
10	2, 7, 8, 9	fvmptd3 6999	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝐹‘𝑥) = ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)))
11		fveq2 6867	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (2nd ‘𝑐) = (2nd ‘𝑦))
12		2fveq3 6872	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑦)))
13	12	oveq1d 7411	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑦)) − 1))
14	11, 13	oveq12d 7414	. . . . . . 7 ⊢ (𝑐 = 𝑦 → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)))
15		id 22	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐶)
16		ovexd 7431	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) ∈ V)
17	2, 14, 15, 16	fvmptd3 6999	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝐹‘𝑦) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)))
18	10, 17	eqeqan12d 2777	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))))
19	18	adantl 485	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))))
20		simplrl 786	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 ∈ 𝐶)
21		simplrr 787	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑦 ∈ 𝐶)
22		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑥) = (1st ‘𝑥)
23		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑥) = (2nd ‘𝑥)
24	1, 22, 23	clwlkclwwlkflem 30213	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐶 → ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ))
25		wlklenvm1 29829	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → (♯‘(1st ‘𝑥)) = ((♯‘(2nd ‘𝑥)) − 1))
26	25	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
27	26	3ad2ant1 1147	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
28	24, 27	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐶 → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
29	28	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
30	29	oveq2d 7412	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))))
31		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑦) = (1st ‘𝑦)
32		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑦) = (2nd ‘𝑦)
33	1, 31, 32	clwlkclwwlkflem 30213	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ))
34		wlklenvm1 29829	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → (♯‘(1st ‘𝑦)) = ((♯‘(2nd ‘𝑦)) − 1))
35	34	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
36	35	3ad2ant1 1147	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
37	33, 36	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐶 → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
38	37	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
39	38	oveq2d 7412	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦))))
40	30, 39	eqeq12d 2779	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) ↔ ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦)))))
41	40	adantl 485	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) ↔ ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦)))))
42	41	biimpa 480	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦))))
43	20, 21, 42	3jca 1142	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦)))))
44	1, 22, 23, 31, 32	clwlkclwwlkf1lem2 30214	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦)))) → ((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖)))
45		simpl 486	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)))
46	43, 44, 45	3syl 18	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)))
47	1, 22, 23, 31, 32	clwlkclwwlkf1lem3 30215	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ∧ ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦)))) → ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))
48	43, 47	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))
49		simpl 486	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
50		wlkcpr 29836	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Walks‘𝐺) ↔ (1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥))
51	50	biimpri 230	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → 𝑥 ∈ (Walks‘𝐺))
52	51	3ad2ant1 1147	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
53	24, 52	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ (Walks‘𝐺))
54		wlkcpr 29836	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (Walks‘𝐺) ↔ (1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦))
55	54	biimpri 230	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → 𝑦 ∈ (Walks‘𝐺))
56	55	3ad2ant1 1147	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
57	33, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ (Walks‘𝐺))
58	53, 57	anim12i 622	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
59	58	adantl 485	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60		eqidd 2764	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥)))
61	49, 59, 60	3jca 1142	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
62	61	adantr 484	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
63		uspgr2wlkeq 29853	. . . . . . 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))))
64	62, 63	syl 17	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → (𝑥 = 𝑦 ↔ ((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0...(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖))))
65	46, 48, 64	mpbir2and 723	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 = 𝑦)
66	65	ex 416	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) → 𝑥 = 𝑦))
67	19, 66	sylbid 242	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
68	67	ralrimivva 3206	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69		dff13 7238	. 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 592	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  {crab 3415  Vcvv 3455   class class class wbr 5101   ↦ cmpt 5182  ⟶wf 6517  –1-1→wf1 6518  ‘cfv 6521  (class class class)co 7396  1^st c1st 7968  2^nd c2nd 7969  0cc0 11084  1c1 11085   ≤ cle 11228   − cmin 11425  ℕcn 12220  ...cfz 13522  ..^cfzo 13669  ♯chash 14353   prefix cpfx 14694  USPGraphcuspgr 29356  Walkscwlks 29804  ClWalkscclwlks 29977  ClWWalkscclwwlk 30190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1075  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8678  df-map 8810  df-pm 8811  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-dju 9871  df-card 9909  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-nn 12221  df-2 12290  df-n0 12492  df-xnn0 12565  df-z 12579  df-uz 12850  df-fz 13523  df-fzo 13670  df-hash 14354  df-word 14537  df-lsw 14586  df-substr 14665  df-pfx 14695  df-edg 29256  df-uhgr 29266  df-upgr 29290  df-uspgr 29358  df-wlks 29807  df-clwlks 29978  df-clwwlk 30191

This theorem is referenced by:  clwlkclwwlkf1o  30220