Theorem wlkl0 30755

Description: There is exactly one walk of length 0 on each vertex 𝑋. (Contributed by AV, 4-Jun-2022.)

Hypothesis

Ref	Expression
clwlknon2num.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wlkl0	⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋

Proof of Theorem wlkl0

Step	Hyp	Ref	Expression
1		clwlkwlk 30161	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 30014	. . . . . . . 8 ⊢ (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
4		fvex 6901	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑤) ∈ V
5		hasheq0 14419	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑤) ∈ V → ((♯‘(1st ‘𝑤)) = 0 ↔ (1st ‘𝑤) = ∅))
6	4, 5	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((♯‘(1st ‘𝑤)) = 0 ↔ (1st ‘𝑤) = ∅)
7	6	birani 509	. . . . . . . . . . . 12 ⊢ (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → (1st ‘𝑤) = ∅)
8	7	3ad2ant3 1153	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (1st ‘𝑤) = ∅)
9	7	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (1st ‘𝑤) = ∅)
10	9	breq1d 5124	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ↔ ∅(ClWalks‘𝐺)(2nd ‘𝑤)))
11		clwlknon2num.v	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑉 = (Vtx‘𝐺)
12	11	1vgrex 29389	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
13	11	0clwlk 30518	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
14	12, 13	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
15	14	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
16	10, 15	bitrd 282	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
17		fz0sn 13674	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
18	17	feq2i 6704	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑤):(0...0)⟶𝑉 ↔ (2nd ‘𝑤):{0}⟶𝑉)
19		c0ex 11218	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
20	19	fsn2 7139	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑤):{0}⟶𝑉 ↔ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩}))
21		simprr 785	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})
22		simprr 785	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((2nd ‘𝑤)‘0) = 𝑋)
23	22	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → ((2nd ‘𝑤)‘0) = 𝑋)
24	23	opeq2d 4850	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → ⟨0, ((2nd ‘𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
25	24	sneqd 4606	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → {⟨0, ((2nd ‘𝑤)‘0)⟩} = {⟨0, 𝑋⟩})
26	21, 25	eqtrd 2801	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
27	26	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩}) → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
28	20, 27	biimtrid 245	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((2nd ‘𝑤):{0}⟶𝑉 → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
29	18, 28	biimtrid 245	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((2nd ‘𝑤):(0...0)⟶𝑉 → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
30	16, 29	sylbid 243	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
31	30	ex 418	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑉 → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})))
32	31	com23 87	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})))
33	32	3imp 1128	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
34	8, 33	opeq12d 4851	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
35	34	3exp 1137	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
36		eleq1 2854	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺)))
37		df-br 5115	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺))
38	36, 37	bitr4di 292	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤)))
39		eqeq1 2770	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
40	39	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → ((((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩) ↔ (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
41	38, 40	imbi12d 347	. . . . . . . . 9 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → ((𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)) ↔ ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))))
42	35, 41	imbitrrid 249	. . . . . . . 8 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑋 ∈ 𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
43	42	com23 87	. . . . . . 7 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) → (𝑋 ∈ 𝑉 → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
44	3, 43	mpcom 39	. . . . . 6 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → (𝑋 ∈ 𝑉 → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
45	44	com12 33	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
46	45	impd 416	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
47		eqidd 2767	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
48	19	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
49		snidg 4631	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
50	48, 49	fsnd 6872	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
51	11	0clwlkv 30519	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
52	47, 50, 51	mpd3an23 1492	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
53		hash0 14423	. . . . . . 7 ⊢ (♯‘∅) = 0
54	53	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
55		fvsng 7185	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
56	19, 55	mpan 703	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
57	52, 54, 56	jca32 525	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
58		eleq1 2854	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
59		df-br 5115	. . . . . . 7 ⊢ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
60	58, 59	bitr4di 292	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
61		0ex 5275	. . . . . . . . 9 ⊢ ∅ ∈ V
62		snex 5415	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
63	61, 62	op1std 8005	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st ‘𝑤) = ∅)
64	63	fveqeq2d 6896	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
65	61, 62	op2ndd 8006	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
66	65	fveq1d 6890	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
67	66	eqeq1d 2768	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd ‘𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
68	64, 67	anbi12d 644	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
69	60, 68	anbi12d 644	. . . . 5 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
70	57, 69	syl5ibrcom 250	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋))))
71	46, 70	impbid 215	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
72	71	alrimiv 1960	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
73		rabeqsn 4638	. 2 ⊢ ({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩} ↔ ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74	72, 73	sylibr 237	1 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458  ∅c0 4289  {csn 4594  ⟨cop 4600   class class class wbr 5114  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423  1^st c1st 7993  2^nd c2nd 7994  0cc0 11118  ...cfz 13553  ♯chash 14386  Vtxcvtx 29383  Walkscwlks 29983  ClWalkscclwlks 30156

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-pm 8836  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-fzo 13702  df-hash 14387  df-word 14571  df-wlks 29986  df-clwlks 30157

This theorem is used by:  numclwlk1lem1  30757