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Theorem wlkl0 30294

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 29703	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 29554	. . . . . . . 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 6888	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑤) ∈ V
5		hasheq0 14379	. . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑤)) = 0 → (1^st ‘𝑤) = ∅)
8	7	adantr 480	. . . . . . . . . . . 12 ⊢ (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → (1^st ‘𝑤) = ∅)
9	8	3ad2ant3 1135	. . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → (1^st ‘𝑤) = ∅)
11	10	breq1d 5129	. . . . . . . . . . . . . . . 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 28927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
14	12	0clwlk 30057	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 13642	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
19	18	feq2i 6697	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑤):(0...0)⟶𝑉 ↔ (2^nd ‘𝑤):{0}⟶𝑉)
20		c0ex 11227	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
21	20	fsn2 7125	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . 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 4856	. . . . . . . . . . . . . . . . . . . 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 4613	. . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 1110	. . . . . . . . . . 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 4857	. . . . . . . . . 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 1119	. . . . . . . . 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 5120	. . . . . . . . . . 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 2739	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
41	40	imbi2d 340	. . . . . . . . . 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 344	. . . . . . . . 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 246	. . . . . . . 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 410	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48		eqidd 2736	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
49	20	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
50		snidg 4636	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
51	49, 50	fsnd 6860	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52	12	0clwlkv 30058	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
53	48, 51, 52	mpd3an23 1465	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54		hash0 14383	. . . . . . 7 ⊢ (♯‘∅) = 0
55	54	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
56		fvsng 7171	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
57	20, 56	mpan 690	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
58	53, 55, 57	jca32 515	. . . . 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 5120	. . . . . . 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 5277	. . . . . . . . 9 ⊢ ∅ ∈ V
63		snex 5406	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
64	62, 63	op1std 7996	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1^st ‘𝑤) = ∅)
65	64	fveqeq2d 6883	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1^st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
66	62, 63	op2ndd 7997	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})
67	66	fveq1d 6877	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2^nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
68	67	eqeq1d 2737	. . . . . . 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 247	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋))))
72	47, 71	impbid 212	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
73	72	alrimiv 1927	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74		rabeqsn 4643	. 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 234	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 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {crab 3415 Vcvv 3459 ∅c0 4308 {csn 4601 ⟨cop 4607 class class class wbr 5119 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 1^st c1st 7984 2^nd c2nd 7985 0cc0 11127 ...cfz 13522 ♯chash 14346 Vtxcvtx 28921 Walkscwlks 29522 ClWalkscclwlks 29698

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-map 8840 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-n0 12500 df-z 12587 df-uz 12851 df-fz 13523 df-fzo 13670 df-hash 14347 df-word 14530 df-wlks 29525 df-clwlks 29699

This theorem is referenced by: numclwlk1lem1 30296

W3C validator