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Theorem wlkl0 28148

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 27558	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 27411	. . . . . . . 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 6685	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘𝑤) ∈ V
5		hasheq0 13727	. . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑤)) = 0 → (1^st ‘𝑤) = ∅)
8	7	adantr 483	. . . . . . . . . . . 12 ⊢ (((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) → (1^st ‘𝑤) = ∅)
9	8	3ad2ant3 1131	. . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → (1^st ‘𝑤) = ∅)
11	10	breq1d 5078	. . . . . . . . . . . . . . . 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 26789	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
14	12	0clwlk 27911	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . 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 13010	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
19	18	feq2i 6508	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑤):(0...0)⟶𝑉 ↔ (2^nd ‘𝑤):{0}⟶𝑉)
20		c0ex 10637	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
21	20	fsn2 6900	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑤):{0}⟶𝑉 ↔ (((2^nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2^nd ‘𝑤) = {⟨0, ((2^nd ‘𝑤)‘0)⟩}))
22		simprr 771	. . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → ((2^nd ‘𝑤)‘0) = 𝑋)
24	23	adantr 483	. . . . . . . . . . . . . . . . . . . . 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 4812	. . . . . . . . . . . . . . . . . . . 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 4581	. . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . . 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 244	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . . 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 1107	. . . . . . . . . . 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 4813	. . . . . . . . . 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 1115	. . . . . . . . 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 2902	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺)))
38		df-br 5069	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤) ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺))
39	37, 38	syl6bbr 291	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1^st ‘𝑤)(ClWalks‘𝐺)(2^nd ‘𝑤)))
40		eqeq1 2827	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
41	40	imbi2d 343	. . . . . . . . . 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 347	. . . . . . . . 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 248	. . . . . . . 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 413	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48		eqidd 2824	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
49	20	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
50		snidg 4601	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
51	49, 50	fsnd 6659	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52	12	0clwlkv 27912	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
53	48, 51, 52	mpd3an23 1459	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54		hash0 13731	. . . . . . 7 ⊢ (♯‘∅) = 0
55	54	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
56		fvsng 6944	. . . . . . 7 ⊢ ((0 ∈ V ∧ 𝑋 ∈ 𝑉) → ({⟨0, 𝑋⟩}‘0) = 𝑋)
57	20, 56	mpan 688	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ({⟨0, 𝑋⟩}‘0) = 𝑋)
58	53, 55, 57	jca32 518	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59		eleq1 2902	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60		df-br 5069	. . . . . . 7 ⊢ (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺))
61	59, 60	syl6bbr 291	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ∅(ClWalks‘𝐺){⟨0, 𝑋⟩}))
62		0ex 5213	. . . . . . . . 9 ⊢ ∅ ∈ V
63		snex 5334	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
64	62, 63	op1std 7701	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1^st ‘𝑤) = ∅)
65	64	fveqeq2d 6680	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1^st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
66	62, 63	op2ndd 7702	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2^nd ‘𝑤) = {⟨0, 𝑋⟩})
67	66	fveq1d 6674	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2^nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
68	67	eqeq1d 2825	. . . . . . 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 249	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋))))
72	47, 71	impbid 214	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
73	72	alrimiv 1928	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74		rabeqsn 4608	. 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 236	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 208 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∈ wcel 2114 {crab 3144 Vcvv 3496 ∅c0 4293 {csn 4569 ⟨cop 4575 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 0cc0 10539 ...cfz 12895 ♯chash 13693 Vtxcvtx 26783 Walkscwlks 27380 ClWalkscclwlks 27553

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-wlks 27383 df-clwlks 27554

This theorem is referenced by: numclwlk1lem1 28150

W3C validator