clwlknf1oclwwlkn - Metamath Proof Explorer

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Theorem clwlknf1oclwwlkn 30028

Description: There is a one-to-one onto function between the set of closed walks as words of length 𝑁 and the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 1-Nov-2022.)

**Hypotheses**
Ref	Expression
clwlknf1oclwwlkn.a	⊢ 𝐴 = (1^st ‘𝑐)
clwlknf1oclwwlkn.b	⊢ 𝐵 = (2^nd ‘𝑐)
clwlknf1oclwwlkn.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}
clwlknf1oclwwlkn.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))

**Assertion**
Ref	Expression
clwlknf1oclwwlkn	⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺))

Distinct variable groups: 𝐶,𝑐 𝐺,𝑐,𝑤 𝑤,𝑁,𝑐 Allowed substitution hints: 𝐴(𝑤,𝑐) 𝐵(𝑤,𝑐) 𝐶(𝑤) 𝐹(𝑤,𝑐)

Proof of Theorem clwlknf1oclwwlkn Dummy variables 𝑑 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
2		2fveq3 6827	. . . . . . . 8 ⊢ (𝑠 = 𝑤 → (♯‘(1^st ‘𝑠)) = (♯‘(1^st ‘𝑤)))
3	2	breq2d 5104	. . . . . . 7 ⊢ (𝑠 = 𝑤 → (1 ≤ (♯‘(1^st ‘𝑠)) ↔ 1 ≤ (♯‘(1^st ‘𝑤))))
4	3	cbvrabv 3405	. . . . . 6 ⊢ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
5		fveq2 6822	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (2^nd ‘𝑑) = (2^nd ‘𝑐))
6		2fveq3 6827	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (♯‘(2^nd ‘𝑑)) = (♯‘(2^nd ‘𝑐)))
7	6	oveq1d 7364	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ((♯‘(2^nd ‘𝑑)) − 1) = ((♯‘(2^nd ‘𝑐)) − 1))
8	5, 7	oveq12d 7367	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
9	8	cbvmptv 5196	. . . . . 6 ⊢ (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
10	4, 9	clwlkclwwlkf1o 29955	. . . . 5 ⊢ (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}–1-1-onto→(ClWWalks‘𝐺))
11	10	adantr 480	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}–1-1-onto→(ClWWalks‘𝐺))
12		2fveq3 6827	. . . . . . . . . 10 ⊢ (𝑤 = 𝑠 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑠)))
13	12	breq2d 5104	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑠))))
14	13	cbvrabv 3405	. . . . . . . 8 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}
15	14	mpteq1i 5183	. . . . . . 7 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
16		fveq2 6822	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (2^nd ‘𝑐) = (2^nd ‘𝑑))
17		2fveq3 6827	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑑)))
18	17	oveq1d 7364	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑑)) − 1))
19	16, 18	oveq12d 7367	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)))
20	19	cbvmptv 5196	. . . . . . 7 ⊢ (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)))
21	15, 20	eqtri 2752	. . . . . 6 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)))
22	21	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))))
23	4	eqcomi 2738	. . . . . 6 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}
24	23	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))})
25		eqidd 2730	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
26	22, 24, 25	f1oeq123d 6758	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}–1-1-onto→(ClWWalks‘𝐺) ↔ (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} ↦ ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}–1-1-onto→(ClWWalks‘𝐺)))
27	11, 26	mpbird 257	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}–1-1-onto→(ClWWalks‘𝐺))
28		fveq2 6822	. . . . . 6 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) → (♯‘𝑠) = (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))))
29	28	3ad2ant3 1135	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∧ 𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) → (♯‘𝑠) = (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))))
30		2fveq3 6827	. . . . . . . . 9 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
31	30	breq2d 5104	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
32	31	elrab 3648	. . . . . . 7 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
33		clwlknf1oclwwlknlem1 30025	. . . . . . 7 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (♯‘(1^st ‘𝑐)))
34	32, 33	sylbi 217	. . . . . 6 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} → (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (♯‘(1^st ‘𝑐)))
35	34	3ad2ant2 1134	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∧ 𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) → (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (♯‘(1^st ‘𝑐)))
36	29, 35	eqtrd 2764	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∧ 𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) → (♯‘𝑠) = (♯‘(1^st ‘𝑐)))
37	36	eqeq1d 2731	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∧ 𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) → ((♯‘𝑠) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
38	1, 27, 37	f1oresrab 7061	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁}):{𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁}–1-1-onto→{𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁})
39		clwlknf1oclwwlkn.a	. . . . 5 ⊢ 𝐴 = (1^st ‘𝑐)
40		clwlknf1oclwwlkn.b	. . . . 5 ⊢ 𝐵 = (2^nd ‘𝑐)
41		clwlknf1oclwwlkn.c	. . . . 5 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁}
42		clwlknf1oclwwlkn.f	. . . . 5 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))
43	39, 40, 41, 42	clwlknf1oclwwlknlem3 30027	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))
44	40	a1i 11	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}) → 𝐵 = (2^nd ‘𝑐))
45		clwlkwlk 29720	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
46		wlkcpr 29574	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
47	39	fveq2i 6825	. . . . . . . . . . . . 13 ⊢ (♯‘𝐴) = (♯‘(1^st ‘𝑐))
48		wlklenvm1 29567	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(1^st ‘𝑐)) = ((♯‘(2^nd ‘𝑐)) − 1))
49	47, 48	eqtrid 2776	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
50	46, 49	sylbi 217	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (Walks‘𝐺) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
51	45, 50	syl 17	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
52	51	adantr 480	. . . . . . . . 9 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
53	32, 52	sylbi 217	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
54	53	adantl 481	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
55	44, 54	oveq12d 7367	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}) → (𝐵 prefix (♯‘𝐴)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
56	55	mpteq2dva 5185	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))))
57	30	eqeq1d 2731	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
58	57	cbvrabv 3405	. . . . . . 7 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁}
59		nnge1 12156	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
60		breq2 5096	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ 𝑁))
61	59, 60	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
62	61	adantl 481	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
63	62	adantr 480	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
64	63	pm4.71rd 562	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
65	64	rabbidva 3401	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
66	58, 65	eqtrid 2776	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
67	32	anbi1i 624	. . . . . . . 8 ⊢ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∧ (♯‘(1^st ‘𝑐)) = 𝑁) ↔ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) = 𝑁))
68		anass 468	. . . . . . . 8 ⊢ (((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
69	67, 68	bitri 275	. . . . . . 7 ⊢ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∧ (♯‘(1^st ‘𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
70	69	rabbia2 3397	. . . . . 6 ⊢ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)}
71	66, 41, 70	3eqtr4g 2789	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
72	56, 71	reseq12d 5931	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁}))
73	43, 72	eqtrd 2764	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁}))
74		clwlknf1oclwwlknlem2 30026	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
75	74	adantl 481	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
76	75, 41, 70	3eqtr4g 2789	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
77		clwwlkn 29970	. . . 4 ⊢ (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁}
78	77	a1i 11	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁})
79	73, 76, 78	f1oeq123d 6758	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺) ↔ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁}):{𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁}–1-1-onto→{𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁}))
80	38, 79	mpbird 257	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 {crab 3394 class class class wbr 5092 ↦ cmpt 5173 ↾ cres 5621 –1-1-onto→wf1o 6481 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 1c1 11010 ≤ cle 11150 − cmin 11347 ℕcn 12128 ♯chash 14237 prefix cpfx 14577 USPGraphcuspgr 29093 Walkscwlks 29542 ClWalkscclwlks 29715 ClWWalkscclwwlk 29925 ClWWalksN cclwwlkn 29968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

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 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14503 df-substr 14548 df-pfx 14578 df-edg 28993 df-uhgr 29003 df-upgr 29027 df-uspgr 29095 df-wlks 29545 df-clwlks 29716 df-clwwlk 29926 df-clwwlkn 29969

This theorem is referenced by: clwlkssizeeq 30029 clwwlknonclwlknonf1o 30306

W3C validator