dlwwlknondlwlknonf1o - Metamath Proof Explorer

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Theorem dlwwlknondlwlknonf1o 28725

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) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
dlwwlknondlwlknonbij.d	⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
dlwwlknondlwlknonf1o.f	⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘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) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
2		df-3an 1088	. . . . 5 ⊢ (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋) ↔ (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋))
3	2	rabbii 3406	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
4	1, 3	eqtri 2768	. . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
5		eqid 2740	. . 3 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}
6		dlwwlknondlwlknonf1o.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
7		eqid 2740	. . 3 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))))
8		eluz2nn 12623	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
9		dlwwlknondlwlknonbij.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
10	9, 5, 7	clwwlknonclwlknonf1o 28722	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
11	8, 10	syl3an3 1164	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))):{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
12		fveq1 6770	. . . . . . 7 ⊢ (𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))) → (𝑦‘(𝑁 − 2)) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)))
13	12	3ad2ant3 1134	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → (𝑦‘(𝑁 − 2)) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)))
14		2fveq3 6776	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
15	14	eqeq1d 2742	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
16		fveq2 6771	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑐 → (2^nd ‘𝑤) = (2^nd ‘𝑐))
17	16	fveq1d 6773	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑐)‘0))
18	17	eqeq1d 2742	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
19	15, 18	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)))
20	19	elrab 3626	. . . . . . . . . 10 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)))
21		simplrl 774	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2))) → (♯‘(1^st ‘𝑐)) = 𝑁)
22		simpll 764	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2))) → 𝑐 ∈ (ClWalks‘𝐺))
23		simpr3 1195	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2))) → 𝑁 ∈ (ℤ_≥‘2))
24	21, 22, 23	3jca 1127	. . . . . . . . . . 11 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2))) → ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)))
25	24	ex 413	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)) → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2))))
26	20, 25	sylbi 216	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} → ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2))))
27	26	impcom 408	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) → ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)))
28		dlwwlknondlwlknonf1olem1 28724	. . . . . . . 8 ⊢ (((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
29	27, 28	syl 17	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
30	29	3adant3 1131	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
31	13, 30	eqtrd 2780	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → (𝑦‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
32	31	eqeq1d 2742	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → ((𝑦‘(𝑁 − 2)) = 𝑋 ↔ ((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋))
33		nfv 1921	. . . . 5 ⊢ Ⅎ𝑤((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋
34	16	fveq1d 6773	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
35	34	eqeq1d 2742	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋))
36	33, 35	sbiev 2313	. . . 4 ⊢ ([𝑐 / 𝑤]((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋)
37	32, 36	bitr4di 289	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ∧ 𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) → ((𝑦‘(𝑁 − 2)) = 𝑋 ↔ [𝑐 / 𝑤]((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋))
38	4, 5, 6, 7, 11, 37	f1ossf1o 6997	. 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 6770	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤‘(𝑁 − 2)) = (𝑦‘(𝑁 − 2)))
41	40	eqeq1d 2742	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑦‘(𝑁 − 2)) = 𝑋))
42	41	cbvrabv 3425	. . . 4 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
43	39, 42	eqtri 2768	. . 3 ⊢ 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
44		f1oeq3 6704	. . 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 233	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝐹:𝑊–1-1-onto→𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1542 [wsb 2071 ∈ wcel 2110 {crab 3070 ↦ cmpt 5162 –1-1-onto→wf1o 6431 ‘cfv 6432 (class class class)co 7271 1^st c1st 7822 2^nd c2nd 7823 0cc0 10872 − cmin 11205 ℕcn 11973 2c2 12028 ℤ_≥cuz 12581 ♯chash 14042 prefix cpfx 14381 Vtxcvtx 27364 USPGraphcuspgr 27516 ClWalkscclwlks 28134 ClWWalksNOncclwwlknon 28447

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-oadd 8292 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-dju 9660 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12582 df-rp 12730 df-fz 13239 df-fzo 13382 df-hash 14043 df-word 14216 df-lsw 14264 df-concat 14272 df-s1 14299 df-substr 14352 df-pfx 14382 df-edg 27416 df-uhgr 27426 df-upgr 27450 df-uspgr 27518 df-wlks 27964 df-clwlks 28135 df-clwwlk 28342 df-clwwlkn 28385 df-clwwlknon 28448

This theorem is referenced by: dlwwlknondlwlknonen 28726

W3C validator