Theorem clwlkclwwlkf1 28374

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 28372	. 2 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
4		fveq2 6774	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → (2nd ‘𝑐) = (2nd ‘𝑥))
5		2fveq3 6779	. . . . . . . . 9 ⊢ (𝑐 = 𝑥 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑥)))
6	5	oveq1d 7290	. . . . . . . 8 ⊢ (𝑐 = 𝑥 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑥)) − 1))
7	4, 6	oveq12d 7293	. . . . . . 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 7310	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) ∈ V)
10	2, 7, 8, 9	fvmptd3 6898	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → (𝐹‘𝑥) = ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)))
11		fveq2 6774	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → (2nd ‘𝑐) = (2nd ‘𝑦))
12		2fveq3 6779	. . . . . . . . 9 ⊢ (𝑐 = 𝑦 → (♯‘(2nd ‘𝑐)) = (♯‘(2nd ‘𝑦)))
13	12	oveq1d 7290	. . . . . . . 8 ⊢ (𝑐 = 𝑦 → ((♯‘(2nd ‘𝑐)) − 1) = ((♯‘(2nd ‘𝑦)) − 1))
14	11, 13	oveq12d 7293	. . . . . . 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 7310	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) ∈ V)
17	2, 14, 15, 16	fvmptd3 6898	. . . . . 6 ⊢ (𝑦 ∈ 𝐶 → (𝐹‘𝑦) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)))
18	10, 17	eqeqan12d 2752	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))))
19	18	adantl 482	. . . 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 774	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 ∈ 𝐶)
21		simplrr 775	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑦 ∈ 𝐶)
22		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑥) = (1st ‘𝑥)
23		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑥) = (2nd ‘𝑥)
24	1, 22, 23	clwlkclwwlkflem 28368	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐶 → ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) ∧ ((2nd ‘𝑥)‘0) = ((2nd ‘𝑥)‘(♯‘(1st ‘𝑥))) ∧ (♯‘(1st ‘𝑥)) ∈ ℕ))
25		wlklenvm1 27989	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → (♯‘(1st ‘𝑥)) = ((♯‘(2nd ‘𝑥)) − 1))
26	25	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
27	26	3ad2ant1 1132	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑥)) − 1) = (♯‘(1st ‘𝑥)))
30	29	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑥) prefix (♯‘(1st ‘𝑥))))
31		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑦) = (1st ‘𝑦)
32		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑦) = (2nd ‘𝑦)
33	1, 31, 32	clwlkclwwlkflem 28368	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐶 → ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) ∧ ((2nd ‘𝑦)‘0) = ((2nd ‘𝑦)‘(♯‘(1st ‘𝑦))) ∧ (♯‘(1st ‘𝑦)) ∈ ℕ))
34		wlklenvm1 27989	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → (♯‘(1st ‘𝑦)) = ((♯‘(2nd ‘𝑦)) − 1))
35	34	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
36	35	3ad2ant1 1132	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((♯‘(2nd ‘𝑦)) − 1) = (♯‘(1st ‘𝑦)))
39	38	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1)) = ((2nd ‘𝑦) prefix (♯‘(1st ‘𝑦))))
40	30, 39	eqeq12d 2754	. . . . . . . . . 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 482	. . . . . . . . 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 477	. . . . . . . 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 1127	. . . . . . 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 28369	. . . . . . 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 483	. . . . . . 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 28370	. . . . . . 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 483	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐺 ∈ USPGraph)
50		wlkcpr 27996	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (Walks‘𝐺) ↔ (1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥))
51	50	biimpri 227	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑥)(Walks‘𝐺)(2nd ‘𝑥) → 𝑥 ∈ (Walks‘𝐺))
52	51	3ad2ant1 1132	. . . . . . . . . . . 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 27996	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (Walks‘𝐺) ↔ (1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦))
55	54	biimpri 227	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑦)(Walks‘𝐺)(2nd ‘𝑦) → 𝑦 ∈ (Walks‘𝐺))
56	55	3ad2ant1 1132	. . . . . . . . . . . 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 613	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
59	58	adantl 482	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)))
60		eqidd 2739	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥)))
61	49, 59, 60	3jca 1127	. . . . . . . 8 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝐺 ∈ USPGraph ∧ (𝑥 ∈ (Walks‘𝐺) ∧ 𝑦 ∈ (Walks‘𝐺)) ∧ (♯‘(1st ‘𝑥)) = (♯‘(1st ‘𝑥))))
62	61	adantr 481	. . . . . . 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 28013	. . . . . . 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 710	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) ∧ ((2nd ‘𝑥) prefix ((♯‘(2nd ‘𝑥)) − 1)) = ((2nd ‘𝑦) prefix ((♯‘(2nd ‘𝑦)) − 1))) → 𝑥 = 𝑦)
66	65	ex 413	. . . 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 239	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
68	67	ralrimivva 3123	. 2 ⊢ (𝐺 ∈ USPGraph → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
69		dff13 7128	. 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 583	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  0cc0 10871  1c1 10872   ≤ cle 11010   − cmin 11205  ℕcn 11973  ...cfz 13239  ..^cfzo 13382  ♯chash 14044   prefix cpfx 14383  USPGraphcuspgr 27518  Walkscwlks 27963  ClWalkscclwlks 28138  ClWWalkscclwwlk 28345

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-lsw 14266  df-substr 14354  df-pfx 14384  df-edg 27418  df-uhgr 27428  df-upgr 27452  df-uspgr 27520  df-wlks 27966  df-clwlks 28139  df-clwwlk 28346

This theorem is referenced by:  clwlkclwwlkf1o  28375