Theorem wlkl0 30855

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 30249	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 30095	. . . . . . . 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 6895	. . . . . . . . . . . . . 14 ⊢ (1st ‘𝑤) ∈ V
5		hasheq0 14431	. . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . 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 29467	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
13	11	0clwlk 30608	. . . . . . . . . . . . . . . . . 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 13686	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
18	17	feq2i 6698	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑤):(0...0)⟶𝑉 ↔ (2nd ‘𝑤):{0}⟶𝑉)
19		c0ex 11228	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
20	19	fsn2 7134	. . . . . . . . . . . . . . . . 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 4843	. . . . . . . . . . . . . . . . . . . 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 4599	. . . . . . . . . . . . . . . . . . 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 2797	. . . . . . . . . . . . . . . . . 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 4844	. . . . . . . . . 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 2850	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺)))
37		df-br 5108	. . . . . . . . . . 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 2766	. . . . . . . . . . 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 2763	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
48	19	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
49		snidg 4624	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
50	48, 49	fsnd 6866	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
51	11	0clwlkv 30609	. . . . . . 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 14435	. . . . . . 7 ⊢ (♯‘∅) = 0
54	53	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
55		fvsng 7182	. . . . . . 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 2850	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
59		df-br 5108	. . . . . . 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 5268	. . . . . . . . 9 ⊢ ∅ ∈ V
62		snex 5408	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
63	61, 62	op1std 8000	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st ‘𝑤) = ∅)
64	63	fveqeq2d 6890	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
65	61, 62	op2ndd 8001	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
66	65	fveq1d 6884	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
67	66	eqeq1d 2764	. . . . . . 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 4631	. 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 2145  {crab 3414  Vcvv 3453  ∅c0 4282  {csn 4587  ⟨cop 4593   class class class wbr 5107  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  0cc0 11128  ...cfz 13565  ♯chash 14398  Vtxcvtx 29461  Walkscwlks 30064  ClWalkscclwlks 30244

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-pm 8833  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-hash 14399  df-word 14583  df-wlks 30067  df-clwlks 30245

This theorem is used by:  numclwlk1lem1  30857