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Theorem clwlknf1oclwwlkn 30103
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 2737 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
2 2fveq3 6911 . . . . . . . 8 (𝑠 = 𝑤 → (♯‘(1st𝑠)) = (♯‘(1st𝑤)))
32breq2d 5155 . . . . . . 7 (𝑠 = 𝑤 → (1 ≤ (♯‘(1st𝑠)) ↔ 1 ≤ (♯‘(1st𝑤))))
43cbvrabv 3447 . . . . . 6 {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
5 fveq2 6906 . . . . . . . 8 (𝑑 = 𝑐 → (2nd𝑑) = (2nd𝑐))
6 2fveq3 6911 . . . . . . . . 9 (𝑑 = 𝑐 → (♯‘(2nd𝑑)) = (♯‘(2nd𝑐)))
76oveq1d 7446 . . . . . . . 8 (𝑑 = 𝑐 → ((♯‘(2nd𝑑)) − 1) = ((♯‘(2nd𝑐)) − 1))
85, 7oveq12d 7449 . . . . . . 7 (𝑑 = 𝑐 → ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
98cbvmptv 5255 . . . . . 6 (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
104, 9clwlkclwwlkf1o 30030 . . . . 5 (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1))):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺))
1110adantr 480 . . . 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 6911 . . . . . . . . . 10 (𝑤 = 𝑠 → (♯‘(1st𝑤)) = (♯‘(1st𝑠)))
1312breq2d 5155 . . . . . . . . 9 (𝑤 = 𝑠 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑠))))
1413cbvrabv 3447 . . . . . . . 8 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}
1514mpteq1i 5238 . . . . . . 7 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
16 fveq2 6906 . . . . . . . . 9 (𝑐 = 𝑑 → (2nd𝑐) = (2nd𝑑))
17 2fveq3 6911 . . . . . . . . . 10 (𝑐 = 𝑑 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑑)))
1817oveq1d 7446 . . . . . . . . 9 (𝑐 = 𝑑 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑑)) − 1))
1916, 18oveq12d 7449 . . . . . . . 8 (𝑐 = 𝑑 → ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) = ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)))
2019cbvmptv 5255 . . . . . . 7 (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) prefix ((♯‘(2nd𝑑)) − 1)))
2115, 20eqtri 2765 . . . . . 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 2746 . . . . . 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 2738 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
2622, 24, 25f1oeq123d 6842 . . . 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 257 . . 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 6906 . . . . . 6 (𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)) → (♯‘𝑠) = (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))))
29283ad2ant3 1136 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → (♯‘𝑠) = (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))))
30 2fveq3 6911 . . . . . . . . 9 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
3130breq2d 5155 . . . . . . . 8 (𝑤 = 𝑐 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑐))))
3231elrab 3692 . . . . . . 7 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))))
33 clwlknf1oclwwlknlem1 30100 . . . . . . 7 ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) → (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (♯‘(1st𝑐)))
3432, 33sylbi 217 . . . . . 6 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (♯‘(1st𝑐)))
35343ad2ant2 1135 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → (♯‘((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) = (♯‘(1st𝑐)))
3629, 35eqtrd 2777 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → (♯‘𝑠) = (♯‘(1st𝑐)))
3736eqeq1d 2739 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1))) → ((♯‘𝑠) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
381, 27, 37f1oresrab 7147 . 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 30102 . . . 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 29795 . . . . . . . . . . 11 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
46 wlkcpr 29647 . . . . . . . . . . . 12 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
4739fveq2i 6909 . . . . . . . . . . . . 13 (♯‘𝐴) = (♯‘(1st𝑐))
48 wlklenvm1 29640 . . . . . . . . . . . . 13 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(1st𝑐)) = ((♯‘(2nd𝑐)) − 1))
4947, 48eqtrid 2789 . . . . . . . . . . . 12 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5046, 49sylbi 217 . . . . . . . . . . 11 (𝑐 ∈ (Walks‘𝐺) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5145, 50syl 17 . . . . . . . . . 10 (𝑐 ∈ (ClWalks‘𝐺) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5251adantr 480 . . . . . . . . 9 ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5332, 52sylbi 217 . . . . . . . 8 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5453adantl 481 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5544, 54oveq12d 7449 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → (𝐵 prefix (♯‘𝐴)) = ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))
5655mpteq2dva 5242 . . . . 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 2739 . . . . . . . 8 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
5857cbvrabv 3447 . . . . . . 7 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}
59 nnge1 12294 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
60 breq2 5147 . . . . . . . . . . . 12 ((♯‘(1st𝑐)) = 𝑁 → (1 ≤ (♯‘(1st𝑐)) ↔ 1 ≤ 𝑁))
6159, 60syl5ibrcom 247 . . . . . . . . . . 11 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6261adantl 481 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6362adantr 480 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6463pm4.71rd 562 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
6564rabbidva 3443 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
6658, 65eqtrid 2789 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
6732anbi1i 624 . . . . . . . 8 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ (♯‘(1st𝑐)) = 𝑁) ↔ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) = 𝑁))
68 anass 468 . . . . . . . 8 (((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
6967, 68bitri 275 . . . . . . 7 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ (♯‘(1st𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
7069rabbia2 3439 . . . . . 6 {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)}
7166, 41, 703eqtr4g 2802 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁})
7256, 71reseq12d 5998 . . . 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 2777 . . 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 30101 . . . . 5 (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7574adantl 481 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7675, 41, 703eqtr4g 2802 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁})
77 clwwlkn 30045 . . . 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 6842 . 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 257 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {crab 3436   class class class wbr 5143  cmpt 5225  cres 5687  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  1c1 11156  cle 11296  cmin 11492  cn 12266  chash 14369   prefix cpfx 14708  USPGraphcuspgr 29165  Walkscwlks 29614  ClWalkscclwlks 29790  ClWWalkscclwwlk 30000   ClWWalksN cclwwlkn 30043
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-edg 29065  df-uhgr 29075  df-upgr 29099  df-uspgr 29167  df-wlks 29617  df-clwlks 29791  df-clwwlk 30001  df-clwwlkn 30044
This theorem is referenced by:  clwlkssizeeq  30104  clwwlknonclwlknonf1o  30381
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