clwlknf1oclwwlkn - Metamath Proof Explorer

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Theorem clwlknf1oclwwlkn 29326

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 2732	. . 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 6893	. . . . . . . 8 ⊢ (𝑠 = 𝑤 → (♯‘(1^st ‘𝑠)) = (♯‘(1^st ‘𝑤)))
3	2	breq2d 5159	. . . . . . 7 ⊢ (𝑠 = 𝑤 → (1 ≤ (♯‘(1^st ‘𝑠)) ↔ 1 ≤ (♯‘(1^st ‘𝑤))))
4	3	cbvrabv 3442	. . . . . 6 ⊢ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
5		fveq2 6888	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (2^nd ‘𝑑) = (2^nd ‘𝑐))
6		2fveq3 6893	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (♯‘(2^nd ‘𝑑)) = (♯‘(2^nd ‘𝑐)))
7	6	oveq1d 7420	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ((♯‘(2^nd ‘𝑑)) − 1) = ((♯‘(2^nd ‘𝑐)) − 1))
8	5, 7	oveq12d 7423	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
9	8	cbvmptv 5260	. . . . . 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 29253	. . . . 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 481	. . . 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 6893	. . . . . . . . . 10 ⊢ (𝑤 = 𝑠 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑠)))
13	12	breq2d 5159	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑠))))
14	13	cbvrabv 3442	. . . . . . . 8 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}
15	14	mpteq1i 5243	. . . . . . 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 6888	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (2^nd ‘𝑐) = (2^nd ‘𝑑))
17		2fveq3 6893	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑑)))
18	17	oveq1d 7420	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑑)) − 1))
19	16, 18	oveq12d 7423	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)))
20	19	cbvmptv 5260	. . . . . . 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 2760	. . . . . 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 2741	. . . . . 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 2733	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
26	22, 24, 25	f1oeq123d 6824	. . . 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 256	. . 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 6888	. . . . . 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 6893	. . . . . . . . 9 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
31	30	breq2d 5159	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
32	31	elrab 3682	. . . . . . 7 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
33		clwlknf1oclwwlknlem1 29323	. . . . . . 7 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (♯‘(1^st ‘𝑐)))
34	32, 33	sylbi 216	. . . . . 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 2772	. . . 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 2734	. . 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 7121	. 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 29325	. . . 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 29021	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
46		wlkcpr 28875	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
47	39	fveq2i 6891	. . . . . . . . . . . . 13 ⊢ (♯‘𝐴) = (♯‘(1^st ‘𝑐))
48		wlklenvm1 28868	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(1^st ‘𝑐)) = ((♯‘(2^nd ‘𝑐)) − 1))
49	47, 48	eqtrid 2784	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
50	46, 49	sylbi 216	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
53	32, 52	sylbi 216	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
54	53	adantl 482	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
55	44, 54	oveq12d 7423	. . . . . 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 5247	. . . . 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 2734	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
58	57	cbvrabv 3442	. . . . . . 7 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁}
59		nnge1 12236	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
60		breq2 5151	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ 𝑁))
61	59, 60	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
62	61	adantl 482	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
63	62	adantr 481	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
64	63	pm4.71rd 563	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
65	64	rabbidva 3439	. . . . . . 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 2784	. . . . . 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 469	. . . . . . . 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 274	. . . . . . 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 3435	. . . . . 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 2797	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
72	56, 71	reseq12d 5980	. . . 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 2772	. . 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 29324	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
75	74	adantl 482	. . . 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 2797	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
77		clwwlkn 29268	. . . 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 6824	. 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 256	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 class class class wbr 5147 ↦ cmpt 5230 ↾ cres 5677 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 1c1 11107 ≤ cle 11245 − cmin 11440 ℕcn 12208 ♯chash 14286 prefix cpfx 14616 USPGraphcuspgr 28397 Walkscwlks 28842 ClWalkscclwlks 29016 ClWWalkscclwwlk 29223 ClWWalksN cclwwlkn 29266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-lsw 14509 df-concat 14517 df-s1 14542 df-substr 14587 df-pfx 14617 df-edg 28297 df-uhgr 28307 df-upgr 28331 df-uspgr 28399 df-wlks 28845 df-clwlks 29017 df-clwwlk 29224 df-clwwlkn 29267

This theorem is referenced by: clwlkssizeeq 29327 clwwlknonclwlknonf1o 29604

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