Theorem clwlkclwwlkf1 30327

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 30325	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (2nd ‘𝑐) = (2nd ‘𝑥))
5		2fveq3 6886	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑥)))
6	5	oveq1d 7425	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑥)) − 1))
7	4, 6	oveq12d 7428	. . . . . . 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 7445	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) ∈ V)
10	2, 7, 8, 9	fvmptd3 7013	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝐹‘𝑥) = ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)))
11		fveq2 6881	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (2nd ‘𝑐) = (2nd ‘𝑦))
12		2fveq3 6886	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑦)))
13	12	oveq1d 7425	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑦)) − 1))
14	11, 13	oveq12d 7428	. . . . . . 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 7445	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) ∈ V)
17	2, 14, 15, 16	fvmptd3 7013	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝐹‘𝑦) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)))
18	10, 17	eqeqan12d 2775	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))))
19	18	adantl 486	. . . 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 788	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 ∈ 𝐶)
21		simplrr 789	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑦 ∈ 𝐶)
22		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑥) = (1st ‘𝑥)
23		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑥) = (2nd ‘𝑥)
24	1, 22, 23	clwlkclwwlkflem 30321	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐶 → ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ))
25		wlklenvm1 29937	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → (♯‘(1st ‘𝑥)) = ((♯‘(2nd ‘𝑥)) − 1))
26	25	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
27	26	3ad2ant1 1149	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
30	29	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))))
31		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑦) = (1st ‘𝑦)
32		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑦) = (2nd ‘𝑦)
33	1, 31, 32	clwlkclwwlkflem 30321	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ))
34		wlklenvm1 29937	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → (♯‘(1st ‘𝑦)) = ((♯‘(2nd ‘𝑦)) − 1))
35	34	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
36	35	3ad2ant1 1149	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
39	38	oveq2d 7426	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦))))
40	30, 39	eqeq12d 2777	. . . . . . . . . 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 486	. . . . . . . . 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 481	. . . . . . . 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 1144	. . . . . . 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 30322	. . . . . . 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 487	. . . . . . 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 30323	. . . . . . 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 487	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
50		wlkcpr 29944	. . . . . . . . . . . . . 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 1149	. . . . . . . . . . . 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 29944	. . . . . . . . . . . . . 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 1149	. . . . . . . . . . . 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 624	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
59	58	adantl 486	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60		eqidd 2762	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥)))
61	49, 59, 60	3jca 1144	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
62	61	adantr 485	. . . . . . 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 29961	. . . . . . 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 725	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 = 𝑦)
66	65	ex 417	. . . 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 3206	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69		dff13 7252	. 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 594	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414  Vcvv 3453   class class class wbr 5108   ↦ cmpt 5191  ⟶wf 6532  –1-1→wf1 6533  ‘cfv 6536  (class class class)co 7410  1^st c1st 7983  2^nd c2nd 7984  0cc0 11099  1c1 11100   ≤ cle 11243   − cmin 11440  ℕcn 12232  ...cfz 13534  ..^cfzo 13681  ♯chash 14365   prefix cpfx 14707  USPGraphcuspgr 29464  Walkscwlks 29912  ClWalkscclwlks 30085  ClWWalkscclwwlk 30298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-xnn0 12577  df-z 12591  df-uz 12862  df-fz 13535  df-fzo 13682  df-hash 14366  df-word 14550  df-lsw 14599  df-substr 14678  df-pfx 14708  df-edg 29364  df-uhgr 29374  df-upgr 29398  df-uspgr 29466  df-wlks 29915  df-clwlks 30086  df-clwwlk 30299

This theorem is referenced by:  clwlkclwwlkf1o  30328