wlksnwwlknvbij - Mathbox for Alexander van der Vekens

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Theorem wlksnwwlknvbij 41114

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 ∈ (1Walks‘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 6113	. . . . 5 ⊢ (1Walks‘𝐺) ∈ V
2	1	mptrabex 6392	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ∈ V
3	2	resex 5363	. . 3 ⊢ ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) ∈ V
4		eqid 2610	. . . 4 ⊢ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) = (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝))
5		usgruspgr 40408	. . . . . 6 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
6		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (1^st ‘𝑞) = (1^st ‘𝑡))
7	6	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → (#‘(1^st ‘𝑞)) = (#‘(1^st ‘𝑡)))
8	7	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑞 = 𝑡 → ((#‘(1^st ‘𝑞)) = 𝑁 ↔ (#‘(1^st ‘𝑡)) = 𝑁))
9	8	cbvrabv 3172	. . . . . . 7 ⊢ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} = {𝑡 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑡)) = 𝑁}
10		eqid 2610	. . . . . . 7 ⊢ (𝑁 WWalkSN 𝐺) = (𝑁 WWalkSN 𝐺)
11		fveq2 6103	. . . . . . . 8 ⊢ (𝑝 = 𝑠 → (2^nd ‘𝑝) = (2^nd ‘𝑠))
12	11	cbvmptv 4678	. . . . . . 7 ⊢ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) = (𝑠 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑠))
13	9, 10, 12	wlknwwlksnbij 41088	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))
14	5, 13	sylan 487	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀) → (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))
15	14	3adant3 1074	. . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)):{𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁}–1-1-onto→(𝑁 WWalkSN 𝐺))
16		fveq1 6102	. . . . . 6 ⊢ (𝑤 = (2^nd ‘𝑝) → (𝑤‘0) = ((2^nd ‘𝑝)‘0))
17	16	eqeq1d 2612	. . . . 5 ⊢ (𝑤 = (2^nd ‘𝑝) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
18	17	3ad2ant3 1077	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) ∧ 𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ 𝑤 = (2^nd ‘𝑝)) → ((𝑤‘0) = 𝑋 ↔ ((2^nd ‘𝑝)‘0) = 𝑋))
19	4, 15, 18	f1oresrab 6302	. . 3 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
20		f1oeq1 6040	. . . 4 ⊢ (𝑓 = ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) → (𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
21	20	spcegv 3267	. . 3 ⊢ (((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}) ∈ V → (((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↦ (2^nd ‘𝑝)) ↾ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}):{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘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 66	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋})
23		df-rab 2905	. . . . 5 ⊢ {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋))}
24		anass 679	. . . . . . 7 ⊢ (((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)))
25	24	bicomi 213	. . . . . 6 ⊢ ((𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)) ↔ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋))
26	25	abbii 2726	. . . . 5 ⊢ {𝑝 ∣ (𝑝 ∈ (1Walks‘𝐺) ∧ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋))} = {𝑝 ∣ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
27		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑝 → (1^st ‘𝑞) = (1^st ‘𝑝))
28	27	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑝 → (#‘(1^st ‘𝑞)) = (#‘(1^st ‘𝑝)))
29	28	eqeq1d 2612	. . . . . . . . . 10 ⊢ (𝑞 = 𝑝 → ((#‘(1^st ‘𝑞)) = 𝑁 ↔ (#‘(1^st ‘𝑝)) = 𝑁))
30	29	elrab 3331	. . . . . . . . 9 ⊢ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ↔ (𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁))
31	30	anbi1i 727	. . . . . . . 8 ⊢ ((𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋) ↔ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋))
32	31	bicomi 213	. . . . . . 7 ⊢ (((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋) ↔ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋))
33	32	abbii 2726	. . . . . 6 ⊢ {𝑝 ∣ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
34		df-rab 2905	. . . . . 6 ⊢ {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋} = {𝑝 ∣ (𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}
35	33, 34	eqtr4i 2635	. . . . 5 ⊢ {𝑝 ∣ ((𝑝 ∈ (1Walks‘𝐺) ∧ (#‘(1^st ‘𝑝)) = 𝑁) ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}
36	23, 26, 35	3eqtri 2636	. . . 4 ⊢ {𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}
37		f1oeq2 6041	. . . 4 ⊢ ({𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)} = {𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋} → (𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘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 ∈ (1Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
39	38	exbidv 1837	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (∃𝑓 𝑓:{𝑝 ∈ (1Walks‘𝐺) ∣ ((#‘(1^st ‘𝑝)) = 𝑁 ∧ ((2^nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋} ↔ ∃𝑓 𝑓:{𝑝 ∈ {𝑞 ∈ (1Walks‘𝐺) ∣ (#‘(1^st ‘𝑞)) = 𝑁} ∣ ((2^nd ‘𝑝)‘0) = 𝑋}–1-1-onto→{𝑤 ∈ (𝑁 WWalkSN 𝐺) ∣ (𝑤‘0) = 𝑋}))
40	22, 39	mpbird 246	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (1Walks‘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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 {crab 2900 Vcvv 3173 ↦ cmpt 4643 ↾ cres 5040 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 ℕ₀cn0 11169 #chash 12979 Vtxcvtx 25673 USPGraph cuspgr 40378 USGraph cusgr 40379 1Walksc1wlks 40796 WWalkSN cwwlksn 41029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-uhgr 25724 df-upgr 25749 df-edga 25793 df-uspgr 40380 df-usgr 40381 df-1wlks 40800 df-wlks 40801 df-wwlks 41033 df-wwlksn 41034

This theorem is referenced by: rusgrnumwlkg 41180

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