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Theorem clwlknf1oclwwlkn 30675

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 2761	. . 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 6890	. . . . . . . 8 ⊢ (𝑠 = 𝑤 → (♯‘(1^st ‘𝑠)) = (♯‘(1^st ‘𝑤)))
3	2	breq2d 5115	. . . . . . 7 ⊢ (𝑠 = 𝑤 → (1 ≤ (♯‘(1^st ‘𝑠)) ↔ 1 ≤ (♯‘(1^st ‘𝑤))))
4	3	cbvrabv 3423	. . . . . 6 ⊢ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
5		fveq2 6885	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (2^nd ‘𝑑) = (2^nd ‘𝑐))
6		2fveq3 6890	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (♯‘(2^nd ‘𝑑)) = (♯‘(2^nd ‘𝑐)))
7	6	oveq1d 7435	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ((♯‘(2^nd ‘𝑑)) − 1) = ((♯‘(2^nd ‘𝑐)) − 1))
8	5, 7	oveq12d 7438	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
9	8	cbvmptv 5209	. . . . . 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 30602	. . . . 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 486	. . . 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 6890	. . . . . . . . . 10 ⊢ (𝑤 = 𝑠 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑠)))
13	12	breq2d 5115	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑠))))
14	13	cbvrabv 3423	. . . . . . . 8 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}
15	14	mpteq1i 5196	. . . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (2^nd ‘𝑐) = (2^nd ‘𝑑))
17		2fveq3 6890	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑑)))
18	17	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑑)) − 1))
19	16, 18	oveq12d 7438	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)))
20	19	cbvmptv 5209	. . . . . . 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 2784	. . . . . 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 2770	. . . . . 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 2762	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
26	22, 24, 25	f1oeq123d 6818	. . . 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 260	. . 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 6885	. . . . . 6 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) → (♯‘𝑠) = (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))))
29	28	3ad2ant3 1153	. . . . 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 6890	. . . . . . . . 9 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
31	30	breq2d 5115	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
32	31	elrab 3645	. . . . . . 7 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
33		clwlknf1oclwwlknlem1 30672	. . . . . . 7 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (♯‘(1^st ‘𝑐)))
34	32, 33	sylbi 220	. . . . . 6 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} → (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))) = (♯‘(1^st ‘𝑐)))
35	34	3ad2ant2 1152	. . . . 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 2796	. . . 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 2763	. . 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 7128	. 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 30674	. . . 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 30362	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
46		wlkcpr 30209	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
47	39	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (♯‘𝐴) = (♯‘(1^st ‘𝑐))
48		wlklenvm1 30202	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(1^st ‘𝑐)) = ((♯‘(2^nd ‘𝑐)) − 1))
49	47, 48	eqtrid 2808	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
50	46, 49	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (Walks‘𝐺) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
51	45, 50	syl 18	. . . . . . . . . 10 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
52	51	adantr 486	. . . . . . . . 9 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
53	32, 52	sylbi 220	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
54	53	adantl 487	. . . . . . 7 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}) → (♯‘𝐴) = ((♯‘(2^nd ‘𝑐)) − 1))
55	44, 54	oveq12d 7438	. . . . . 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 5198	. . . . 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 2763	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
58	57	cbvrabv 3423	. . . . . . 7 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁}
59		nnge1 12366	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
60		breq2 5107	. . . . . . . . . . . 12 ⊢ ((♯‘(1^st ‘𝑐)) = 𝑁 → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ 𝑁))
61	59, 60	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
62	61	adantl 487	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
63	62	adantr 486	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 → 1 ≤ (♯‘(1^st ‘𝑐))))
64	63	pm4.71rd 572	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1^st ‘𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)))
65	64	rabbidva 3419	. . . . . . 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 2808	. . . . . 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 636	. . . . . . . 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 474	. . . . . . . 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 278	. . . . . . 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 3416	. . . . . 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 2821	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
72	56, 71	reseq12d 5971	. . . 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 2796	. . 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 30673	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1^st ‘𝑐)) ∧ (♯‘(1^st ‘𝑐)) = 𝑁)})
75	74	adantl 487	. . . 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 2821	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
77		clwwlkn 30617	. . . 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 6818	. 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 260	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 {crab 3413 class class class wbr 5103 ↦ cmpt 5186 ↾ cres 5653 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 1c1 11201 ≤ cle 11344 − cmin 11541 ℕcn 12335 ♯chash 14474 prefix cpfx 14820 USPGraphcuspgr 29729 Walkscwlks 30177 ClWalkscclwlks 30357 ClWWalkscclwwlk 30572 ClWWalksN cclwwlkn 30615

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-oadd 8480 df-er 8717 df-map 8849 df-pm 8850 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-n0 12607 df-xnn0 12680 df-z 12694 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-lsw 14708 df-concat 14716 df-s1 14743 df-substr 14789 df-pfx 14821 df-edg 29626 df-uhgr 29636 df-upgr 29660 df-uspgr 29731 df-wlks 30180 df-clwlks 30358 df-clwwlk 30573 df-clwwlkn 30616

This theorem is used by: clwlkssizeeq 30676 clwwlknonclwlknonf1o 30963

W3C validator