dlwwlknondlwlknonf1o - Metamath Proof Explorer

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Theorem dlwwlknondlwlknonf1o 28128

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 1085	. . . . 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 3465	. . . 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 2844	. . 3 ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)}
5		eqid 2821	. . 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 2821	. . 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 12271	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℕ)
9		dlwwlknondlwlknonbij.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
10	9, 5, 7	clwwlknonclwlknonf1o 28125	. . . 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 1161	. . 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 6655	. . . . . . 7 ⊢ (𝑦 = ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐))) → (𝑦‘(𝑁 − 2)) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)))
13	12	3ad2ant3 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)))) → (𝑦‘(𝑁 − 2)) = (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)))
14		2fveq3 6661	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
15	14	eqeq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
16		fveq2 6656	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑐 → (2^nd ‘𝑤) = (2^nd ‘𝑐))
17	16	fveq1d 6658	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑐)‘0))
18	17	eqeq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘0) = 𝑋 ↔ ((2^nd ‘𝑐)‘0) = 𝑋))
19	15, 18	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑐 → (((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋) ↔ ((♯‘(1^st ‘𝑐)) = 𝑁 ∧ ((2^nd ‘𝑐)‘0) = 𝑋)))
20	19	elrab 3671	. . . . . . . . . 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 775	. . . . . . . . . . . 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 765	. . . . . . . . . . . 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 1192	. . . . . . . . . . . 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 1124	. . . . . . . . . . 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 415	. . . . . . . . . 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 219	. . . . . . . . 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 410	. . . . . . . 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 28127	. . . . . . . 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 1128	. . . . . 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 2856	. . . . 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 2823	. . . 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 1915	. . . . 5 ⊢ Ⅎ𝑤((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋
34	16	fveq1d 6658	. . . . . 6 ⊢ (𝑤 = 𝑐 → ((2^nd ‘𝑤)‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))
35	34	eqeq1d 2823	. . . . 5 ⊢ (𝑤 = 𝑐 → (((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋))
36	33, 35	sbiev 2330	. . . 4 ⊢ ([𝑐 / 𝑤]((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2^nd ‘𝑐)‘(𝑁 − 2)) = 𝑋)
37	32, 36	syl6bbr 291	. . 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 6876	. 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 6655	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤‘(𝑁 − 2)) = (𝑦‘(𝑁 − 2)))
41	40	eqeq1d 2823	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤‘(𝑁 − 2)) = 𝑋 ↔ (𝑦‘(𝑁 − 2)) = 𝑋))
42	41	cbvrabv 3483	. . . 4 ⊢ {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
43	39, 42	eqtri 2844	. . 3 ⊢ 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑦‘(𝑁 − 2)) = 𝑋}
44		f1oeq3 6592	. . 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 236	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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 [wsb 2069 ∈ wcel 2114 {crab 3142 ↦ cmpt 5132 –1-1-onto→wf1o 6340 ‘cfv 6341 (class class class)co 7142 1^st c1st 7673 2^nd c2nd 7674 0cc0 10523 − cmin 10856 ℕcn 11624 2c2 11679 ℤ_≥cuz 12230 ♯chash 13680 prefix cpfx 14017 Vtxcvtx 26767 USPGraphcuspgr 26919 ClWalkscclwlks 27537 ClWWalksNOncclwwlknon 27850

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-dju 9316 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-n0 11885 df-xnn0 11955 df-z 11969 df-uz 12231 df-rp 12377 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-lsw 13900 df-concat 13908 df-s1 13935 df-substr 13988 df-pfx 14018 df-edg 26819 df-uhgr 26829 df-upgr 26853 df-uspgr 26921 df-wlks 27367 df-clwlks 27538 df-clwwlk 27745 df-clwwlkn 27788 df-clwwlknon 27851

This theorem is referenced by: dlwwlknondlwlknonen 28129

W3C validator