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Theorem wlkl0 29620

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) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

Distinct variable groups: 𝑤,𝐺 𝑤,𝑉 𝑤,𝑋

**Proof of Theorem wlkl0**
Step	Hyp	Ref	Expression
1		clwlkwlk 29032	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 28885	. . . . . . . 8 ⊢ (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
4		fvex 6905	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑤) ∈ V
5		hasheq0 14323	. . . . . . . . . . . . . . 15 ⊢ ((1^st ‘𝑤) ∈ V → ((♯‘(1^st ‘𝑤)) = 0 ↔ (1^st ‘𝑤) = ∅))
6	4, 5	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑤)) = 0 ↔ (1^st ‘𝑤) = ∅)
7	6	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑤)) = 0 → (1^st ‘𝑤) = ∅)
8	7	adantr 482	. . . . . . . . . . . 12 ⊢ (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → (1^st ‘𝑤) = ∅)
9	8	3ad2ant3 1136	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ (1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → (1^st ‘𝑤) = ∅)
10	8	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → (1^st ‘𝑤) = ∅)
11	10	breq1d 5159	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ ∅(ClWalks‘𝐺)(2^nd ‘𝑤)))
12		clwlknon2num.v	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑉 = (Vtx‘𝐺)
13	12	1vgrex 28262	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
14	12	0clwlk 29383	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ (2^nd ‘𝑤):(0...0)⟶𝑉))
15	13, 14	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ (2^nd ‘𝑤):(0...0)⟶𝑉))
16	15	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ (2^nd ‘𝑤):(0...0)⟶𝑉))
17	11, 16	bitrd 279	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ (2^nd ‘𝑤):(0...0)⟶𝑉))
18		fz0sn 13601	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
19	18	feq2i 6710	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑤):(0...0)⟶𝑉 ↔ (2^nd ‘𝑤):{0}⟶𝑉)
20		c0ex 11208	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
21	20	fsn2 7134	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑤):{0}⟶𝑉 ↔ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩}))
22		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ∧ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩})) → (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩})
23		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((2^nd ‘𝑤)‘0) = 𝑋)
24	23	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ∧ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩})) → ((2^nd ‘𝑤)‘0) = 𝑋)
25	24	opeq2d 4881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ∧ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩})) → ⟨0, ((2^nd ‘𝑤)‘0)⟩ = ⟨0, 𝑋⟩)
26	25	sneqd 4641	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ∧ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩})) → {⟨0, ((2^nd ‘𝑤)‘0)⟩} = {⟨0, 𝑋⟩})
27	22, 26	eqtrd 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ∧ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩})) → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})
28	27	ex 414	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩}) → (2^nd ‘𝑤) = {⟨0, 𝑋⟩}))
29	21, 28	biimtrid 241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((2^nd ‘𝑤):{0}⟶𝑉 → (2^nd ‘𝑤) = {⟨0, 𝑋⟩}))
30	19, 29	biimtrid 241	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((2^nd ‘𝑤):(0...0)⟶𝑉 → (2^nd ‘𝑤) = {⟨0, 𝑋⟩}))
31	17, 30	sylbid 239	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) → (2^nd ‘𝑤) = {⟨0, 𝑋⟩}))
32	31	ex 414	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑉 → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})))
33	32	com23 86	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})))
34	33	3imp 1112	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ (1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})
35	9, 34	opeq12d 4882	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ (1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
36	35	3exp 1120	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
37		eleq1 2822	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺)))
38		df-br 5150	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺))
39	37, 38	bitr4di 289	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤)))
40		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
41	40	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → ((((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩) ↔ (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
42	39, 41	imbi12d 345	. . . . . . . . 9 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → ((𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)) ↔ ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))))
43	36, 42	imbitrrid 245	. . . . . . . 8 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑋 ∈ 𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
44	43	com23 86	. . . . . . 7 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) → (𝑋 ∈ 𝑉 → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
45	3, 44	mpcom 38	. . . . . 6 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → (𝑋 ∈ 𝑉 → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
46	45	com12 32	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
47	46	impd 412	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48		eqidd 2734	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
49	20	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
50		snidg 4663	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
51	49, 50	fsnd 6877	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52	12	0clwlkv 29384	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
53	48, 51, 52	mpd3an23 1464	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54		hash0 14327	. . . . . . 7 ⊢ (♯‘∅) = 0
55	54	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
56		fvsng 7178	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
57	20, 56	mpan 689	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
58	53, 55, 57	jca32 517	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59		eleq1 2822	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60		df-br 5150	. . . . . . 7 ⊢ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
61	59, 60	bitr4di 289	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62		0ex 5308	. . . . . . . . 9 ⊢ ∅ ∈ V
63		snex 5432	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
64	62, 63	op1std 7985	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1^st ‘𝑤) = ∅)
65	64	fveqeq2d 6900	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1^st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
66	62, 63	op2ndd 7986	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})
67	66	fveq1d 6894	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2^nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
68	67	eqeq1d 2735	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
69	65, 68	anbi12d 632	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
70	61, 69	anbi12d 632	. . . . 5 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋))))
71	58, 70	syl5ibrcom 246	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋))))
72	47, 71	impbid 211	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
73	72	alrimiv 1931	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74		rabeqsn 4670	. 2 ⊢ ({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩} ↔ ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
75	73, 74	sylibr 233	1 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 ∅c0 4323 {csn 4629 ⟨cop 4635 class class class wbr 5149 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 0cc0 11110 ...cfz 13484 ♯chash 14290 Vtxcvtx 28256 Walkscwlks 28853 ClWalkscclwlks 29027

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-wlks 28856 df-clwlks 29028

This theorem is referenced by: numclwlk1lem1 29622

W3C validator