wlksnwwlknvbij - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wlksnwwlknvbij		Structured version Visualization version GIF version

Theorem wlksnwwlknvbij 26803

Description: There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 20-Apr-2021.)

**Assertion**
Ref	Expression
wlksnwwlknvbij	⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

Distinct variable groups: 𝑓,𝐺,𝑝,𝑤 𝑓,𝑁,𝑝,𝑤 𝑓,𝑋,𝑝,𝑤

Proof of Theorem wlksnwwlknvbij Dummy variables 𝑞 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . . 5 ⊢ (Walks‘𝐺) ∈ V
2	1	mptrabex 6488	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ∈ V
3	2	resex 5443	. . 3 ⊢ ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) ∈ V
4		eqid 2622	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) = (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝))
5		usgruspgr 26073	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
6		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (1^st ‘𝑞) = (1^st ‘𝑡))
7	6	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → (#‘(1^st ‘𝑞)) = (#‘(1^st ‘𝑡)))
8	7	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑞 = 𝑡 → ((#‘(1^st ‘𝑞)) = 𝑁 ↔ (#‘(1^st ‘𝑡)) = 𝑁))
9	8	cbvrabv 3199	. . . . . . 7 ⊢ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} = {𝑡 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑡)) = 𝑁}
10		eqid 2622	. . . . . . 7 ⊢ (𝑁 WWalksN 𝐺) = (𝑁 WWalksN 𝐺)
11		fveq2 6191	. . . . . . . 8 ⊢ (𝑝 = 𝑠 → (2^nd ‘𝑝) = (2^nd ‘𝑠))
12	11	cbvmptv 4750	. . . . . . 7 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) = (𝑠 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑠))
13	9, 10, 12	wlknwwlksnbij 26777	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
14	5, 13	sylan 488	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
15	14	3adant3 1081	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))
16		fveq1 6190	. . . . . 6 ⊢ (𝑤 = (2^nd ‘𝑝) → (𝑤‘0) = ((2^nd ‘𝑝)‘0))
17	16	eqeq1d 2624	. . . . 5 ⊢ (𝑤 = (2^nd ‘𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
18	17	3ad2ant3 1084	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) ∧ 𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ 𝑤 = (2^nd ‘𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
19	4, 15, 18	f1oresrab 6395	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
20		f1oeq1 6127	. . . 4 ⊢ (𝑓 = ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) → (𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
21	20	spcegv 3294	. . 3 ⊢ (((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) ∈ V → (((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
22	3, 19, 21	mpsyl 68	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
23		df-rab 2921	. . . . 5 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋))}
24		anass 681	. . . . . . 7 ⊢ (((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)))
25	24	bicomi 214	. . . . . 6 ⊢ ((𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)) ↔ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋))
26	25	abbii 2739	. . . . 5 ⊢ {𝑝 ∣ (𝑝 ∈ (Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋))} = {𝑝 ∣ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
27		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑝 → (1^st ‘𝑞) = (1^st ‘𝑝))
28	27	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑝 → (#‘(1^st ‘𝑞)) = (#‘(1^st ‘𝑝)))
29	28	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑞 = 𝑝 → ((#‘(1^st ‘𝑞)) = 𝑁 ↔ (#‘(1^st ‘𝑝)) = 𝑁))
30	29	elrab 3363	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↔ (𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁))
31	30	anbi1i 731	. . . . . . . 8 ⊢ ((𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋) ↔ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋))
32	31	bicomi 214	. . . . . . 7 ⊢ (((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋))
33	32	abbii 2739	. . . . . 6 ⊢ {𝑝 ∣ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
34		df-rab 2921	. . . . . 6 ⊢ {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
35	33, 34	eqtr4i 2647	. . . . 5 ⊢ {𝑝 ∣ ((𝑝 ∈ (Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}
36	23, 26, 35	3eqtri 2648	. . . 4 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}
37		f1oeq2 6128	. . . 4 ⊢ ({𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋} → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
38	36, 37	mp1i 13	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
39	38	exbidv 1850	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}))
40	22, 39	mpbird 247	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 {cab 2608 {crab 2916 Vcvv 3200 ↦ cmpt 4729 ↾ cres 5116 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 0cc0 9936 ℕ₀cn0 11292 #chash 13117 Vtxcvtx 25874 USPGraph cuspgr 26043 USGraph cusgr 26044 Walkscwlks 26492 WWalksN cwwlksn 26718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-uspgr 26045 df-usgr 26046 df-wlks 26495 df-wwlks 26722 df-wwlksn 26723

This theorem is referenced by: rusgrnumwlkg 26872

W3C validator