Theorem wlkl0 29311

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 28723	. . . . . . . 8 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 ∈ (Walks‘𝐺))
2		wlkop 28576	. . . . . . . 8 ⊢ (𝑤 ∈ (Walks‘𝐺) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → 𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩)
4		fvex 6855	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑤) ∈ V
5		hasheq0 14263	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑤) ∈ V → ((♯‘(1st ‘𝑤)) = 0 ↔ (1st ‘𝑤) = ∅))
6	4, 5	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1st ‘𝑤)) = 0 ↔ (1st ‘𝑤) = ∅)
7	6	biimpi 215	. . . . . . . . . . . . 13 ⊢ ((♯‘(1st ‘𝑤)) = 0 → (1st ‘𝑤) = ∅)
8	7	adantr 481	. . . . . . . . . . . 12 ⊢ (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → (1st ‘𝑤) = ∅)
9	8	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (1st ‘𝑤) = ∅)
10	8	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (1st ‘𝑤) = ∅)
11	10	breq1d 5115	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ↔ ∅(ClWalks‘𝐺)(2nd ‘𝑤)))
12		clwlknon2num.v	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑉 = (Vtx‘𝐺)
13	12	1vgrex 27953	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ 𝑉 → 𝐺 ∈ V)
14	12	0clwlk 29074	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ V → (∅(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
15	13, 14	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
16	15	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (∅(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
17	11, 16	bitrd 278	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ↔ (2nd ‘𝑤):(0...0)⟶𝑉))
18		fz0sn 13541	. . . . . . . . . . . . . . . . 17 ⊢ (0...0) = {0}
19	18	feq2i 6660	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑤):(0...0)⟶𝑉 ↔ (2nd ‘𝑤):{0}⟶𝑉)
20		c0ex 11149	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ V
21	20	fsn2 7082	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑤):{0}⟶𝑉 ↔ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩}))
22		simprr 771	. . . . . . . . . . . . . . . . . . 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)⟩})
23		simprr 771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((2nd ‘𝑤)‘0) = 𝑋)
24	23	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → ((2nd ‘𝑤)‘0) = 𝑋)
25	24	opeq2d 4837	. . . . . . . . . . . . . . . . . . . 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⟩)
26	25	sneqd 4598	. . . . . . . . . . . . . . . . . . 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⟩})
27	22, 26	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ∧ (((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩})) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
28	27	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((((2nd ‘𝑤)‘0) ∈ 𝑉 ∧ (2nd ‘𝑤) = {⟨0, ((2nd ‘𝑤)‘0)⟩}) → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
29	21, 28	biimtrid 241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((2nd ‘𝑤):{0}⟶𝑉 → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
30	19, 29	biimtrid 241	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((2nd ‘𝑤):(0...0)⟶𝑉 → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
31	17, 30	sylbid 239	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝑉 ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) = {⟨0, 𝑋⟩}))
32	31	ex 413	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑉 → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})))
33	32	com23 86	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑉 → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})))
34	33	3imp 1111	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
35	9, 34	opeq12d 4838	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)
36	35	3exp 1119	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑉 → ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩)))
37		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺)))
38		df-br 5106	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤) ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ ∈ (ClWalks‘𝐺))
39	37, 38	bitr4di 288	. . . . . . . . . 10 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ (1st ‘𝑤)(ClWalks‘𝐺)(2nd ‘𝑤)))
40		eqeq1 2740	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ ↔ ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ = ⟨∅, {⟨0, 𝑋⟩}⟩))
41	40	imbi2d 340	. . . . . . . . . 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⟩}⟩)))
42	39, 41	imbi12d 344	. . . . . . . . 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⟩}⟩))))
43	36, 42	syl5ibr 245	. . . . . . . 8 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑋 ∈ 𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
44	43	com23 86	. . . . . . 7 ⊢ (𝑤 = ⟨(1st ‘𝑤), (2nd ‘𝑤)⟩ → (𝑤 ∈ (ClWalks‘𝐺) → (𝑋 ∈ 𝑉 → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))))
45	3, 44	mpcom 38	. . . . . 6 ⊢ (𝑤 ∈ (ClWalks‘𝐺) → (𝑋 ∈ 𝑉 → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
46	45	com12 32	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (𝑤 ∈ (ClWalks‘𝐺) → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩)))
47	46	impd 411	. . . 4 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) → 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
48		eqidd 2737	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∅ = ∅)
49	20	a1i 11	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 0 ∈ V)
50		snidg 4620	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
51	49, 50	fsnd 6827	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → {⟨0, 𝑋⟩}:{0}⟶{𝑋})
52	12	0clwlkv 29075	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∅ = ∅ ∧ {⟨0, 𝑋⟩}:{0}⟶{𝑋}) → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
53	48, 51, 52	mpd3an23 1463	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∅(ClWalks‘𝐺){⟨0, 𝑋⟩})
54		hash0 14267	. . . . . . 7 ⊢ (♯‘∅) = 0
55	54	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (♯‘∅) = 0)
56		fvsng 7126	. . . . . . 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 516	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∅(ClWalks‘𝐺){⟨0, 𝑋⟩} ∧ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
59		eleq1 2825	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (𝑤 ∈ (ClWalks‘𝐺) ↔ ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ (ClWalks‘𝐺)))
60		df-br 5106	. . . . . . 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 5264	. . . . . . . . 9 ⊢ ∅ ∈ V
63		snex 5388	. . . . . . . . 9 ⊢ {⟨0, 𝑋⟩} ∈ V
64	62, 63	op1std 7931	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (1st ‘𝑤) = ∅)
65	64	fveqeq2d 6850	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((♯‘(1st ‘𝑤)) = 0 ↔ (♯‘∅) = 0))
66	62, 63	op2ndd 7932	. . . . . . . . 9 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (2nd ‘𝑤) = {⟨0, 𝑋⟩})
67	66	fveq1d 6844	. . . . . . . 8 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((2nd ‘𝑤)‘0) = ({⟨0, 𝑋⟩}‘0))
68	67	eqeq1d 2738	. . . . . . 7 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((2nd ‘𝑤)‘0) = 𝑋 ↔ ({⟨0, 𝑋⟩}‘0) = 𝑋))
69	65, 68	anbi12d 631	. . . . . 6 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → (((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘∅) = 0 ∧ ({⟨0, 𝑋⟩}‘0) = 𝑋)))
70	61, 69	anbi12d 631	. . . . 5 ⊢ (𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩ → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘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) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋))))
72	47, 71	impbid 211	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
73	72	alrimiv 1930	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑤((𝑤 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)) ↔ 𝑤 = ⟨∅, {⟨0, 𝑋⟩}⟩))
74		rabeqsn 4627	. 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⟩}⟩))
75	73, 74	sylibr 233	1 ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 0 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {crab 3407  Vcvv 3445  ∅c0 4282  {csn 4586  ⟨cop 4592   class class class wbr 5105  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920  0cc0 11051  ...cfz 13424  ♯chash 14230  Vtxcvtx 27947  Walkscwlks 28544  ClWalkscclwlks 28718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-wlks 28547  df-clwlks 28719

This theorem is referenced by:  numclwlk1lem1  29313