Theorem dlwwlknondlwlknonf1o 30389

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 1088	. . . . 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 3402	. . . 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 2757	. . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ∧ ((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
5		eqid 2734	. . 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 2734	. . 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 12799	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ)
9		dlwwlknondlwlknonbij.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
10	9, 5, 7	clwwlknonclwlknonf1o 30386	. . . 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 1165	. . 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 6831	. . . . . . 7 ⊢ (𝑦 = ((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐))) → (𝑦‘(𝑁 − 2)) = (((2nd ‘𝑐) prefix (♯‘(1st ‘𝑐)))‘(𝑁 − 2)))
13	12	3ad2ant3 1135	. . . . . 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 6837	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → (♯‘(1st ‘𝑤)) = (♯‘(1st ‘𝑐)))
15	14	eqeq1d 2736	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → ((♯‘(1st ‘𝑤)) = 𝑁 ↔ (♯‘(1st ‘𝑐)) = 𝑁))
16		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑐 → (2nd ‘𝑤) = (2nd ‘𝑐))
17	16	fveq1d 6834	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → ((2nd ‘𝑤)‘0) = ((2nd ‘𝑐)‘0))
18	17	eqeq1d 2736	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (((2nd ‘𝑤)‘0) = 𝑋 ↔ ((2nd ‘𝑐)‘0) = 𝑋))
19	15, 18	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → (((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)))
20	19	elrab 3644	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑋)} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)))
21		simplrl 776	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → (♯‘(1st ‘𝑐)) = 𝑁)
22		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1st ‘𝑐)) = 𝑁 ∧ ((2nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2))) → 𝑐 ∈ (ClWalks‘𝐺))
23		simpr3 1197	. . . . . . . . . . . 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 1128	. . . . . . . . . . 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 412	. . . . . . . . . 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 217	. . . . . . . . 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 407	. . . . . . . 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 30388	. . . . . . . 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 17	. . . . . . 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 1132	. . . . . 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 2769	. . . . 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 2736	. . . 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 1915	. . . . 5 ⊢ Ⅎ𝑤((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋
34	16	fveq1d 6834	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2nd ‘𝑤)‘(𝑁 − 2)) = ((2nd ‘𝑐)‘(𝑁 − 2)))
35	34	eqeq1d 2736	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋))
36	33, 35	sbiev 2317	. . . 4 ⊢ ([𝑐 / 𝑤]((2nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd ‘𝑐)‘(𝑁 − 2)) = 𝑋)
37	32, 36	bitr4di 289	. . 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 7071	. 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 6831	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤‘(𝑁 − 2)) = (𝑦‘(𝑁 − 2)))
41	40	eqeq1d 2736	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑦‘(𝑁 − 2)) = 𝑋))
42	41	cbvrabv 3407	. . . 4 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
43	39, 42	eqtri 2757	. . 3 ⊢ 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
44		f1oeq3 6762	. . 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 234	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) → 𝐹:𝑊–1-1-onto→𝐷)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  [wsb 2067   ∈ wcel 2113  {crab 3397   ↦ cmpt 5177  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  0cc0 11024   − cmin 11362  ℕcn 12143  2c2 12198  ℤ_≥cuz 12749  ♯chash 14251   prefix cpfx 14592  Vtxcvtx 29018  USPGraphcuspgr 29170  ClWalkscclwlks 29792  ClWWalksNOncclwwlknon 30111

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-n0 12400  df-xnn0 12473  df-z 12487  df-uz 12750  df-rp 12904  df-fz 13422  df-fzo 13569  df-hash 14252  df-word 14435  df-lsw 14484  df-concat 14492  df-s1 14518  df-substr 14563  df-pfx 14593  df-edg 29070  df-uhgr 29080  df-upgr 29104  df-uspgr 29172  df-wlks 29622  df-clwlks 29793  df-clwwlk 30006  df-clwwlkn 30049  df-clwwlknon 30112

This theorem is referenced by:  dlwwlknondlwlknonen  30390