Theorem dlwwlknondlwlknonf1o 30656

Description: 𝐹 is a bijection between the two representations of double loops of a fixed positive length on a fixed vertex. (Contributed by AV, 30-May-2022.) (Revised by AV, 1-Nov-2022.)

Hypotheses

Ref	Expression
dlwwlknondlwlknonbij.v	⊢ 𝑉 = (Vtx‘𝐺)
dlwwlknondlwlknonbij.w	⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
dlwwlknondlwlknonbij.d	⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
dlwwlknondlwlknonf1o.f	⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐))))

Assertion

Ref	Expression
dlwwlknondlwlknonf1o	⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝐹:𝑊–1-1-onto→𝐷)

Distinct variable groups:   𝐺,𝑐,𝑤   𝑁,𝑐,𝑤   𝑉,𝑐   𝑊,𝑐   𝑋,𝑐,𝑤

Allowed substitution hints:   𝐷(𝑤,𝑐)   𝐹(𝑤,𝑐)   𝑉(𝑤)   𝑊(𝑤)

Proof of Theorem dlwwlknondlwlknonf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dlwwlknondlwlknonbij.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
2		df-3an 1103	. . . . 5 ⊢ (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋) ↔ (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋))
3	2	rabbii 3428	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋 ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
4	1, 3	eqtri 2792	. . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
5		eqid 2769	. . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}
6		dlwwlknondlwlknonf1o.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐))))
7		eqid 2769	. . 3 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐))))
8		eluz2nn 12911	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ)
9		dlwwlknondlwlknonbij.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
10	9, 5, 7	clwwlknonclwlknonf1o 30653	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
11	8, 10	syl3an3 1181	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ↦ ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
12		fveq1 6881	. . . . . . 7 ⊢ (𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐))) → (𝑦‘(𝑁 − 2)) = (((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))‘(𝑁 − 2)))
13	12	3ad2ant3 1151	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) → (𝑦‘(𝑁 − 2)) = (((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))‘(𝑁 − 2)))
14		2fveq3 6887	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑐)))
15	14	eqeq1d 2771	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → ((♯‘(1st ‘𝑤)) = 𝑁 ↔ (♯‘(1st ‘𝑐)) = 𝑁))
16		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑐 → (2nd ‘𝑤) = (2nd ‘𝑐))
17	16	fveq1d 6884	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → ((2nd ‘𝑤)‘0) = ((2nd ‘𝑐)‘0))
18	17	eqeq1d 2771	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (((2nd ‘𝑤)‘0) = 𝑋 ↔ ((2nd ‘𝑐)‘0) = 𝑋))
19	15, 18	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)))
20	19	elrab 3659	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)))
21		simplrl 788	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → (♯‘(1st ‘𝑐)) = 𝑁)
22		simpll 778	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → 𝑐 ∈ (ClWalks‘𝐺))
23		simpr3 1213	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → 𝑁 ∈ (ℤ≥‘2))
24	21, 22, 23	3jca 1144	. . . . . . . . . . 11 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → ((♯‘(1st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ≥‘2)))
25	24	ex 417	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → ((♯‘(1st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ≥‘2))))
26	20, 25	sylbi 220	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → ((♯‘(1st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ≥‘2))))
27	26	impcom 412	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) → ((♯‘(1st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ≥‘2)))
28		dlwwlknondlwlknonf1olem1 30655	. . . . . . . 8 ⊢ (((♯‘(1st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ≥‘2)) → (((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))‘(𝑁 − 2)) = ((2nd ‘𝑐)‘(𝑁 − 2)))
29	27, 28	syl 18	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)}) → (((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))‘(𝑁 − 2)) = ((2nd ‘𝑐)‘(𝑁 − 2)))
30	29	3adant3 1148	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) → (((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))‘(𝑁 − 2)) = ((2nd ‘𝑐)‘(𝑁 − 2)))
31	13, 30	eqtrd 2804	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) → (𝑦‘(𝑁 − 2)) = ((2nd ‘𝑐)‘(𝑁 − 2)))
32	31	eqeq1d 2771	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) → ((𝑦‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋))
33		nfv 1941	. . . . 5 ⊢ Ⅎ𝑤((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋
34	16	fveq1d 6884	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2nd ‘𝑤)‘(𝑁 − 2)) = ((2nd ‘𝑐)‘(𝑁 − 2)))
35	34	eqeq1d 2771	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋))
36	33, 35	sbiev 2353	. . . 4 ⊢ ([𝑐 / 𝑤]((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋)
37	32, 36	bitr4di 292	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))) → ((𝑦‘(𝑁 − 2)) = 𝑋 ↔ [𝑐 / 𝑤]((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋))
38	4, 5, 6, 7, 11, 37	f1ossf1o 7125	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝐹:𝑊–1-1-onto→{𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋})
39		dlwwlknondlwlknonbij.d	. . . 4 ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
40		fveq1 6881	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤‘(𝑁 − 2)) = (𝑦‘(𝑁 − 2)))
41	40	eqeq1d 2771	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑦‘(𝑁 − 2)) = 𝑋))
42	41	cbvrabv 3433	. . . 4 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
43	39, 42	eqtri 2792	. . 3 ⊢ 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
44		f1oeq3 6811	. . 3 ⊢ (𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋} → (𝐹:𝑊–1-1-onto→𝐷 ↔ 𝐹:𝑊–1-1-onto→{𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}))
45	43, 44	ax-mp 5	. 2 ⊢ (𝐹:𝑊–1-1-onto→𝐷 ↔ 𝐹:𝑊–1-1-onto→{𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋})
46	38, 45	sylibr 237	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝐹:𝑊–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  [wsb 2097   ∈ wcel 2149  {crab 3423   ↦ cmpt 5196  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  1^st c1st 7983  2^nd c2nd 7984  0cc0 11099   − cmin 11440  ℕcn 12232  2c2 12294  ℤ_≥cuz 12861  ♯chash 14365   prefix cpfx 14707  Vtxcvtx 29286  USPGraphcuspgr 29438  ClWalkscclwlks 30059  ClWWalksNOncclwwlknon 30378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ifp 1077  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-nn 12233  df-2 12302  df-n0 12504  df-xnn0 12577  df-z 12591  df-uz 12862  df-rp 13016  df-fz 13535  df-fzo 13682  df-hash 14366  df-word 14550  df-lsw 14599  df-concat 14607  df-s1 14633  df-substr 14678  df-pfx 14708  df-edg 29338  df-uhgr 29348  df-upgr 29372  df-uspgr 29440  df-wlks 29889  df-clwlks 30060  df-clwwlk 30273  df-clwwlkn 30316  df-clwwlknon 30379

This theorem is referenced by:  dlwwlknondlwlknonen  30657