clwlknf1oclwwlkn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > clwlknf1oclwwlkn		Structured version Visualization version GIF version

Theorem clwlknf1oclwwlkn 28448

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 2738	. . 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 6779	. . . . . . . 8 ⊢ (𝑠 = 𝑤 → (♯‘(1^st ‘𝑠)) = (♯‘(1^st ‘𝑤)))
3	2	breq2d 5086	. . . . . . 7 ⊢ (𝑠 = 𝑤 → (1 ≤ (♯‘(1^st ‘𝑠)) ↔ 1 ≤ (♯‘(1^st ‘𝑤))))
4	3	cbvrabv 3426	. . . . . 6 ⊢ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
5		fveq2 6774	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → (2^nd ‘𝑑) = (2^nd ‘𝑐))
6		2fveq3 6779	. . . . . . . . 9 ⊢ (𝑑 = 𝑐 → (♯‘(2^nd ‘𝑑)) = (♯‘(2^nd ‘𝑐)))
7	6	oveq1d 7290	. . . . . . . 8 ⊢ (𝑑 = 𝑐 → ((♯‘(2^nd ‘𝑑)) − 1) = ((♯‘(2^nd ‘𝑐)) − 1))
8	5, 7	oveq12d 7293	. . . . . . 7 ⊢ (𝑑 = 𝑐 → ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)) = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
9	8	cbvmptv 5187	. . . . . 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 28375	. . . . 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 6779	. . . . . . . . . 10 ⊢ (𝑤 = 𝑠 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑠)))
13	12	breq2d 5086	. . . . . . . . 9 ⊢ (𝑤 = 𝑠 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑠))))
14	13	cbvrabv 3426	. . . . . . . 8 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑠))}
15	14	mpteq1i 5170	. . . . . . 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 6774	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (2^nd ‘𝑐) = (2^nd ‘𝑑))
17		2fveq3 6779	. . . . . . . . . 10 ⊢ (𝑐 = 𝑑 → (♯‘(2^nd ‘𝑐)) = (♯‘(2^nd ‘𝑑)))
18	17	oveq1d 7290	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → ((♯‘(2^nd ‘𝑐)) − 1) = ((♯‘(2^nd ‘𝑑)) − 1))
19	16, 18	oveq12d 7293	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) = ((2^nd ‘𝑑) prefix ((♯‘(2^nd ‘𝑑)) − 1)))
20	19	cbvmptv 5187	. . . . . . 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 2766	. . . . . 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 2747	. . . . . 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 2739	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
26	22, 24, 25	f1oeq123d 6710	. . . 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 6774	. . . . . 6 ⊢ (𝑠 = ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) → (♯‘𝑠) = (♯‘((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1))))
29	28	3ad2ant3 1134	. . . . 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 6779	. . . . . . . . 9 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
31	30	breq2d 5086	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
32	31	elrab 3624	. . . . . . 7 ⊢ (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
33		clwlknf1oclwwlknlem1 28445	. . . . . . 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 1133	. . . . 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 2778	. . . 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 2740	. . 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 6999	. 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 28447	. . . 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 28143	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
46		wlkcpr 27996	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐))
47	39	fveq2i 6777	. . . . . . . . . . . . 13 ⊢ (♯‘𝐴) = (♯‘(1^st ‘𝑐))
48		wlklenvm1 27989	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(1^st ‘𝑐)) = ((♯‘(2^nd ‘𝑐)) − 1))
49	47, 48	eqtrid 2790	. . . . . . . . . . . 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 7293	. . . . . 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 5174	. . . . 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 2740	. . . . . . . 8 ⊢ (𝑤 = 𝑐 → ((♯‘(1^st ‘𝑤)) = 𝑁 ↔ (♯‘(1^st ‘𝑐)) = 𝑁))
58	57	cbvrabv 3426	. . . . . . 7 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1^st ‘𝑐)) = 𝑁}
59		nnge1 12001	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 1 ≤ 𝑁)
60		breq2 5078	. . . . . . . . . . . 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 3413	. . . . . . 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 2790	. . . . . 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 3412	. . . . . 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 2803	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
72	56, 71	reseq12d 5892	. . . 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 2778	. . 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 28446	. . . . 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 2803	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} ∣ (♯‘(1^st ‘𝑐)) = 𝑁})
77		clwwlkn 28390	. . . 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 6710	. 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 1086 = wceq 1539 ∈ wcel 2106 {crab 3068 class class class wbr 5074 ↦ cmpt 5157 ↾ cres 5591 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 1c1 10872 ≤ cle 11010 − cmin 11205 ℕcn 11973 ♯chash 14044 prefix cpfx 14383 USPGraphcuspgr 27518 Walkscwlks 27963 ClWalkscclwlks 28138 ClWWalkscclwwlk 28345 ClWWalksN cclwwlkn 28388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

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 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-edg 27418 df-uhgr 27428 df-upgr 27452 df-uspgr 27520 df-wlks 27966 df-clwlks 28139 df-clwwlk 28346 df-clwwlkn 28389

This theorem is referenced by: clwlkssizeeq 28449 clwwlknonclwlknonf1o 28726

W3C validator