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Theorem wlkl0 28632

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 28044	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 27897	. . . . . . . 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 6769	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑤) ∈ V
5		hasheq0 14006	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → (1^st ‘𝑤) = ∅)
9	8	3ad2ant3 1133	. . . . . . . . . . 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 5080	. . . . . . . . . . . . . . . 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 27275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
14	12	0clwlk 28395	. . . . . . . . . . . . . . . . . 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 278	. . . . . . . . . . . . . . 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 13285	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
19	18	feq2i 6576	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑤):(0...0)⟶𝑉 ↔ (2^nd ‘𝑤):{0}⟶𝑉)
20		c0ex 10900	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
21	20	fsn2 6990	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑤):{0}⟶𝑉 ↔ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩}))
22		simprr 769	. . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . 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 4808	. . . . . . . . . . . . . . . . . . . 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 4570	. . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . 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	syl5bi 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	syl5bi 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 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 1109	. . . . . . . . . . 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 4809	. . . . . . . . . 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 1117	. . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺)))
38		df-br 5071	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺))
39	37, 38	bitr4di 288	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤)))
40		eqeq1 2742	. . . . . . . . . . 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	syl5ibr 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 410	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48		eqidd 2739	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
49	20	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
50		snidg 4592	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
51	49, 50	fsnd 6742	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52	12	0clwlkv 28396	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
53	48, 51, 52	mpd3an23 1461	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54		hash0 14010	. . . . . . 7 ⊢ (♯‘∅) = 0
55	54	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
56		fvsng 7034	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
57	20, 56	mpan 686	. . . . . 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 2826	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60		df-br 5071	. . . . . . 7 ⊢ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
61	59, 60	bitr4di 288	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62		0ex 5226	. . . . . . . . 9 ⊢ ∅ ∈ V
63		snex 5349	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
64	62, 63	op1std 7814	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1^st ‘𝑤) = ∅)
65	64	fveqeq2d 6764	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1^st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
66	62, 63	op2ndd 7815	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})
67	66	fveq1d 6758	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2^nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
68	67	eqeq1d 2740	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
69	65, 68	anbi12d 630	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
70	61, 69	anbi12d 630	. . . . 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 4599	. 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 395 ∧ w3a 1085 ∀wal 1537 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 ∅c0 4253 {csn 4558 ⟨cop 4564 class class class wbr 5070 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 0cc0 10802 ...cfz 13168 ♯chash 13972 Vtxcvtx 27269 Walkscwlks 27866 ClWalkscclwlks 28039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-wlks 27869 df-clwlks 28040

This theorem is referenced by: numclwlk1lem1 28634

W3C validator