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Theorem clwlknf1oclwwlkn 27538
 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 = (1st𝑐)
clwlknf1oclwwlkn.b 𝐵 = (2nd𝑐)
clwlknf1oclwwlkn.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
clwlknf1oclwwlkn.f 𝐹 = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴)))
Assertion
Ref Expression
clwlknf1oclwwlkn ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶1-1-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.
StepHypRef Expression
1 eqid 2793 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
2 2fveq3 6535 . . . . . . . 8 (𝑠 = 𝑤 → (♯‘(1st𝑠)) = (♯‘(1st𝑤)))
32breq2d 4968 . . . . . . 7 (𝑠 = 𝑤 → (1 ≤ (♯‘(1st𝑠)) ↔ 1 ≤ (♯‘(1st𝑤))))
43cbvrabv 3429 . . . . . 6 {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
5 fveq2 6530 . . . . . . . 8 (𝑑 = 𝑐 → (2nd𝑑) = (2nd𝑐))
6 2fveq3 6535 . . . . . . . . 9 (𝑑 = 𝑐 → (♯‘(2nd𝑑)) = (♯‘(2nd𝑐)))
76oveq1d 7022 . . . . . . . 8 (𝑑 = 𝑐 → ((♯‘(2nd𝑑)) − 1) = ((♯‘(2nd𝑐)) − 1))
85, 7oveq12d 7025 . . . . . . 7 (𝑑 = 𝑐 → ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
98cbvmptv 5055 . . . . . 6 (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
104, 9clwlkclwwlkf1o 27464 . . . . 5 (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺))
1110adantr 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺))
12 2fveq3 6535 . . . . . . . . . 10 (𝑤 = 𝑠 → (♯‘(1st𝑤)) = (♯‘(1st𝑠)))
1312breq2d 4968 . . . . . . . . 9 (𝑤 = 𝑠 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑠))))
1413cbvrabv 3429 . . . . . . . 8 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}
1514mpteq1i 5044 . . . . . . 7 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
16 fveq2 6530 . . . . . . . . 9 (𝑐 = 𝑑 → (2nd𝑐) = (2nd𝑑))
17 2fveq3 6535 . . . . . . . . . 10 (𝑐 = 𝑑 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑑)))
1817oveq1d 7022 . . . . . . . . 9 (𝑐 = 𝑑 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑑)) − 1))
1916, 18oveq12d 7025 . . . . . . . 8 (𝑐 = 𝑑 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)))
2019cbvmptv 5055 . . . . . . 7 (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)))
2115, 20eqtri 2817 . . . . . 6 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)))
2221a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))))
234eqcomi 2802 . . . . . 6 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}
2423a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))})
25 eqidd 2794 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
2622, 24, 25f1oeq123d 6470 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}–1-1-onto→(ClWWalks‘𝐺) ↔ (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺)))
2711, 26mpbird 258 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}–1-1-onto→(ClWWalks‘𝐺))
28 fveq2 6530 . . . . . 6 (𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) → (♯‘𝑠) = (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))))
29283ad2ant3 1126 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → (♯‘𝑠) = (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))))
30 2fveq3 6535 . . . . . . . . 9 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
3130breq2d 4968 . . . . . . . 8 (𝑤 = 𝑐 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑐))))
3231elrab 3613 . . . . . . 7 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))))
33 clwlknf1oclwwlknlem1 27535 . . . . . . 7 ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) → (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (♯‘(1st𝑐)))
3432, 33sylbi 218 . . . . . 6 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (♯‘(1st𝑐)))
35343ad2ant2 1125 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (♯‘(1st𝑐)))
3629, 35eqtrd 2829 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → (♯‘𝑠) = (♯‘(1st𝑐)))
3736eqeq1d 2795 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → ((♯‘𝑠) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
381, 27, 37f1oresrab 6743 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}):{𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}–1-1-onto→{𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁})
39 clwlknf1oclwwlkn.a . . . . 5 𝐴 = (1st𝑐)
40 clwlknf1oclwwlkn.b . . . . 5 𝐵 = (2nd𝑐)
41 clwlknf1oclwwlkn.c . . . . 5 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
42 clwlknf1oclwwlkn.f . . . . 5 𝐹 = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴)))
4339, 40, 41, 42clwlknf1oclwwlknlem3 27537 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))
4440a1i 11 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → 𝐵 = (2nd𝑐))
45 clwlkwlk 27231 . . . . . . . . . . 11 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
46 wlkcpr 27081 . . . . . . . . . . . 12 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
4739fveq2i 6533 . . . . . . . . . . . . 13 (♯‘𝐴) = (♯‘(1st𝑐))
48 wlklenvm1 27074 . . . . . . . . . . . . 13 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(1st𝑐)) = ((♯‘(2nd𝑐)) − 1))
4947, 48syl5eq 2841 . . . . . . . . . . . 12 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5046, 49sylbi 218 . . . . . . . . . . 11 (𝑐 ∈ (Walks‘𝐺) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5145, 50syl 17 . . . . . . . . . 10 (𝑐 ∈ (ClWalks‘𝐺) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5251adantr 481 . . . . . . . . 9 ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5332, 52sylbi 218 . . . . . . . 8 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5453adantl 482 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5544, 54oveq12d 7025 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → (𝐵 prefix (♯‘𝐴)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
5655mpteq2dva 5049 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))))
5730eqeq1d 2795 . . . . . . . 8 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
5857cbvrabv 3429 . . . . . . 7 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}
59 nnge1 11502 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
60 breq2 4960 . . . . . . . . . . . 12 ((♯‘(1st𝑐)) = 𝑁 → (1 ≤ (♯‘(1st𝑐)) ↔ 1 ≤ 𝑁))
6159, 60syl5ibrcom 248 . . . . . . . . . . 11 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6261adantl 482 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6362adantr 481 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6463pm4.71rd 563 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
6564rabbidva 3419 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
6658, 65syl5eq 2841 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
6732anbi1i 623 . . . . . . . 8 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ (♯‘(1st𝑐)) = 𝑁) ↔ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) = 𝑁))
68 anass 469 . . . . . . . 8 (((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
6967, 68bitri 276 . . . . . . 7 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ (♯‘(1st𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
7069rabbia2 3418 . . . . . 6 {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)}
7166, 41, 703eqtr4g 2854 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁})
7256, 71reseq12d 5727 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶) = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}))
7343, 72eqtrd 2829 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}))
74 clwlknf1oclwwlknlem2 27536 . . . . 5 (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7574adantl 482 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7675, 41, 703eqtr4g 2854 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁})
77 clwwlkn 27479 . . . 4 (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁}
7877a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁})
7973, 76, 78f1oeq123d 6470 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺) ↔ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}):{𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}–1-1-onto→{𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁}))
8038, 79mpbird 258 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  {crab 3107   class class class wbr 4956   ↦ cmpt 5035   ↾ cres 5437  –1-1-onto→wf1o 6216  ‘cfv 6217  (class class class)co 7007  1st c1st 7534  2nd c2nd 7535  1c1 10373   ≤ cle 10511   − cmin 10706  ℕcn 11475  ♯chash 13528   prefix cpfx 13856  USPGraphcuspgr 26604  Walkscwlks 27049  ClWalkscclwlks 27226  ClWWalkscclwwlk 27434   ClWWalksN cclwwlkn 27477 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ifp 1054  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-oadd 7948  df-er 8130  df-map 8249  df-pm 8250  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-dju 9165  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-2 11537  df-n0 11735  df-xnn0 11805  df-z 11819  df-uz 12083  df-rp 12229  df-fz 12732  df-fzo 12873  df-hash 13529  df-word 13696  df-lsw 13749  df-concat 13757  df-s1 13782  df-substr 13827  df-pfx 13857  df-edg 26504  df-uhgr 26514  df-upgr 26538  df-uspgr 26606  df-wlks 27052  df-clwlks 27227  df-clwwlk 27435  df-clwwlkn 27478 This theorem is referenced by:  clwlkssizeeq  27539  clwwlknonclwlknonf1o  27821
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