Theorem clwlkclwwlkf1 30534

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 30532	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		fveq2 6873	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (2nd ‘𝑐) = (2nd ‘𝑥))
5		2fveq3 6878	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑥)))
6	5	oveq1d 7423	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑥)) − 1))
7	4, 6	oveq12d 7426	. . . . . . 7 ⊢ (𝑐 = 𝑥 → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)))
8		id 23	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
9		ovexd 7443	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) ∈ V)
10	2, 7, 8, 9	fvmptd3 7005	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝐹‘𝑥) = ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)))
11		fveq2 6873	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (2nd ‘𝑐) = (2nd ‘𝑦))
12		2fveq3 6878	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑦)))
13	12	oveq1d 7423	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑦)) − 1))
14	11, 13	oveq12d 7426	. . . . . . 7 ⊢ (𝑐 = 𝑦 → ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)))
15		id 23	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐶)
16		ovexd 7443	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) ∈ V)
17	2, 14, 15, 16	fvmptd3 7005	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝐹‘𝑦) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)))
18	10, 17	eqeqan12d 2774	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))))
19	18	adantl 487	. . . 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 789	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 ∈ 𝐶)
21		simplrr 790	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑦 ∈ 𝐶)
22		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑥) = (1st ‘𝑥)
23		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑥) = (2nd ‘𝑥)
24	1, 22, 23	clwlkclwwlkflem 30528	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐶 → ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ))
25		wlklenvm1 30135	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → (♯‘(1st ‘𝑥)) = ((♯‘(2nd ‘𝑥)) − 1))
26	25	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
27	26	3ad2ant1 1151	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐶 → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
29	28	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
30	29	oveq2d 7424	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))))
31		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑦) = (1st ‘𝑦)
32		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑦) = (2nd ‘𝑦)
33	1, 31, 32	clwlkclwwlkflem 30528	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ))
34		wlklenvm1 30135	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → (♯‘(1st ‘𝑦)) = ((♯‘(2nd ‘𝑦)) − 1))
35	34	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
36	35	3ad2ant1 1151	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐶 → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
38	37	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
39	38	oveq2d 7424	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦))))
40	30, 39	eqeq12d 2776	. . . . . . . . . 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 487	. . . . . . . . 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 482	. . . . . . . 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 1146	. . . . . . 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 30529	. . . . . . 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 488	. . . . . . 7 ⊢ (((♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)) ∧ ∀𝑖 ∈ (0..^(♯‘(1st ‘𝑥)))((2nd ‘𝑥)‘𝑖) = ((2nd ‘𝑦)‘𝑖)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑦)))
46	43, 44, 45	3syl 19	. . . . . 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 30530	. . . . . . 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 18	. . . . . 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 488	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
50		wlkcpr 30142	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Walks‘𝐺) ↔ (1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥))
51	50	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → 𝑥 ∈ (Walks‘𝐺))
52	51	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ) → 𝑥 ∈ (Walks‘𝐺))
53	24, 52	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ (Walks‘𝐺))
54		wlkcpr 30142	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (Walks‘𝐺) ↔ (1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦))
55	54	biimpri 231	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → 𝑦 ∈ (Walks‘𝐺))
56	55	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ) → 𝑦 ∈ (Walks‘𝐺))
57	33, 56	syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝐶 → 𝑦 ∈ (Walks‘𝐺))
58	53, 57	anim12i 625	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
59	58	adantl 487	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60		eqidd 2761	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥)))
61	49, 59, 60	3jca 1146	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
62	61	adantr 486	. . . . . . 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 30159	. . . . . . 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 18	. . . . . 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 726	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 = 𝑦)
66	65	ex 418	. . . 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 243	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
68	67	ralrimivva 3205	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69		dff13 7246	. 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 595	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   class class class wbr 5102   ↦ cmpt 5185  ⟶wf 6523  –1-1→wf1 6524  ‘cfv 6527  (class class class)co 7408  1^st c1st 7982  2^nd c2nd 7983  0cc0 11171  1c1 11172   ≤ cle 11315   − cmin 11512  ℕcn 12304  ...cfz 13608  ..^cfzo 13756  ♯chash 14441   prefix cpfx 14787  USPGraphcuspgr 29662  Walkscwlks 30110  ClWalkscclwlks 30290  ClWWalkscclwwlk 30505

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  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 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-pm 8828  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-dju 9953  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-n0 12576  df-xnn0 12649  df-z 12663  df-uz 12935  df-fz 13609  df-fzo 13757  df-hash 14442  df-word 14626  df-lsw 14675  df-substr 14756  df-pfx 14788  df-edg 29559  df-uhgr 29569  df-upgr 29593  df-uspgr 29664  df-wlks 30113  df-clwlks 30291  df-clwwlk 30506

This theorem is used by:  clwlkclwwlkf1o  30535