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Theorem clwlknf1oclwwlknOLD 27410
Description: Obsolete version of clwlknf1oclwwlkn 27408 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
clwlknf1oclwwlknOLD.a 𝐴 = (1st𝑐)
clwlknf1oclwwlknOLD.b 𝐵 = (2nd𝑐)
clwlknf1oclwwlknOLD.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
clwlknf1oclwwlknOLD.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
Assertion
Ref Expression
clwlknf1oclwwlknOLD ((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 clwlknf1oclwwlknOLD
Dummy variables 𝑑 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2797 . . 3 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
2 2fveq3 6414 . . . . . . . 8 (𝑠 = 𝑤 → (♯‘(1st𝑠)) = (♯‘(1st𝑤)))
32breq2d 4853 . . . . . . 7 (𝑠 = 𝑤 → (1 ≤ (♯‘(1st𝑠)) ↔ 1 ≤ (♯‘(1st𝑤))))
43cbvrabv 3381 . . . . . 6 {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
5 fveq2 6409 . . . . . . . 8 (𝑑 = 𝑐 → (2nd𝑑) = (2nd𝑐))
6 2fveq3 6414 . . . . . . . . . 10 (𝑑 = 𝑐 → (♯‘(2nd𝑑)) = (♯‘(2nd𝑐)))
76oveq1d 6891 . . . . . . . . 9 (𝑑 = 𝑐 → ((♯‘(2nd𝑑)) − 1) = ((♯‘(2nd𝑐)) − 1))
87opeq2d 4598 . . . . . . . 8 (𝑑 = 𝑐 → ⟨0, ((♯‘(2nd𝑑)) − 1)⟩ = ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)
95, 8oveq12d 6894 . . . . . . 7 (𝑑 = 𝑐 → ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩) = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
109cbvmptv 4941 . . . . . 6 (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩)) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
114, 10clwlkclwwlkf1oOLD 27300 . . . . 5 (𝐺 ∈ USPGraph → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩)):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺))
1211adantr 473 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩)):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺))
13 2fveq3 6414 . . . . . . . . . 10 (𝑤 = 𝑠 → (♯‘(1st𝑤)) = (♯‘(1st𝑠)))
1413breq2d 4853 . . . . . . . . 9 (𝑤 = 𝑠 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑠))))
1514cbvrabv 3381 . . . . . . . 8 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}
1615mpteq1i 4930 . . . . . . 7 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
17 fveq2 6409 . . . . . . . . 9 (𝑐 = 𝑑 → (2nd𝑐) = (2nd𝑑))
18 2fveq3 6414 . . . . . . . . . . 11 (𝑐 = 𝑑 → (♯‘(2nd𝑐)) = (♯‘(2nd𝑑)))
1918oveq1d 6891 . . . . . . . . . 10 (𝑐 = 𝑑 → ((♯‘(2nd𝑐)) − 1) = ((♯‘(2nd𝑑)) − 1))
2019opeq2d 4598 . . . . . . . . 9 (𝑐 = 𝑑 → ⟨0, ((♯‘(2nd𝑐)) − 1)⟩ = ⟨0, ((♯‘(2nd𝑑)) − 1)⟩)
2117, 20oveq12d 6894 . . . . . . . 8 (𝑐 = 𝑑 → ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) = ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩))
2221cbvmptv 4941 . . . . . . 7 (𝑐 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩))
2316, 22eqtri 2819 . . . . . 6 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩))
2423a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩)))
254eqcomi 2806 . . . . . 6 {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}
2625a1i 11 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} = {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))})
27 eqidd 2798 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (ClWWalks‘𝐺) = (ClWWalks‘𝐺))
2824, 26, 27f1oeq123d 6349 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}–1-1-onto→(ClWWalks‘𝐺) ↔ (𝑑 ∈ {𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))} ↦ ((2nd𝑑) substr ⟨0, ((♯‘(2nd𝑑)) − 1)⟩)):{𝑠 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑠))}–1-1-onto→(ClWWalks‘𝐺)))
2912, 28mpbird 249 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)):{𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}–1-1-onto→(ClWWalks‘𝐺))
30 fveq2 6409 . . . . . 6 (𝑠 = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩) → (♯‘𝑠) = (♯‘((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)))
31303ad2ant3 1166 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) → (♯‘𝑠) = (♯‘((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)))
32 2fveq3 6414 . . . . . . . . 9 (𝑤 = 𝑐 → (♯‘(1st𝑤)) = (♯‘(1st𝑐)))
3332breq2d 4853 . . . . . . . 8 (𝑤 = 𝑐 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘(1st𝑐))))
3433elrab 3554 . . . . . . 7 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))))
35 clwlknf1oclwwlknlem1OLD 27405 . . . . . . 7 ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) → (♯‘((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (♯‘(1st𝑐)))
3634, 35sylbi 209 . . . . . 6 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → (♯‘((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (♯‘(1st𝑐)))
37363ad2ant2 1165 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) → (♯‘((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) = (♯‘(1st𝑐)))
3831, 37eqtrd 2831 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) → (♯‘𝑠) = (♯‘(1st𝑐)))
3938eqeq1d 2799 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ 𝑠 = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) → ((♯‘𝑠) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
401, 29, 39f1oresrab 6619 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(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})
41 clwlknf1oclwwlknOLD.a . . . . 5 𝐴 = (1st𝑐)
42 clwlknf1oclwwlknOLD.b . . . . 5 𝐵 = (2nd𝑐)
43 clwlknf1oclwwlknOLD.c . . . . 5 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}
44 clwlknf1oclwwlknOLD.f . . . . 5 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
4541, 42, 43, 44clwlknf1oclwwlknlem3OLD 27409 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶))
4642a1i 11 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → 𝐵 = (2nd𝑐))
47 clwlkwlk 27021 . . . . . . . . . . . 12 (𝑐 ∈ (ClWalks‘𝐺) → 𝑐 ∈ (Walks‘𝐺))
48 wlkcpr 26870 . . . . . . . . . . . . 13 (𝑐 ∈ (Walks‘𝐺) ↔ (1st𝑐)(Walks‘𝐺)(2nd𝑐))
4941fveq2i 6412 . . . . . . . . . . . . . 14 (♯‘𝐴) = (♯‘(1st𝑐))
50 wlklenvm1 26863 . . . . . . . . . . . . . 14 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘(1st𝑐)) = ((♯‘(2nd𝑐)) − 1))
5149, 50syl5eq 2843 . . . . . . . . . . . . 13 ((1st𝑐)(Walks‘𝐺)(2nd𝑐) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5248, 51sylbi 209 . . . . . . . . . . . 12 (𝑐 ∈ (Walks‘𝐺) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5347, 52syl 17 . . . . . . . . . . 11 (𝑐 ∈ (ClWalks‘𝐺) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5453adantr 473 . . . . . . . . . 10 ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5534, 54sylbi 209 . . . . . . . . 9 (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5655adantl 474 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → (♯‘𝐴) = ((♯‘(2nd𝑐)) − 1))
5756opeq2d 4598 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → ⟨0, (♯‘𝐴)⟩ = ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)
5846, 57oveq12d 6894 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}) → (𝐵 substr ⟨0, (♯‘𝐴)⟩) = ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))
5958mpteq2dva 4935 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) = (𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)))
6032eqeq1d 2799 . . . . . . . 8 (𝑤 = 𝑐 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁))
6160cbvrabv 3381 . . . . . . 7 {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁}
62 nnge1 11340 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 1 ≤ 𝑁)
63 breq2 4845 . . . . . . . . . . . 12 ((♯‘(1st𝑐)) = 𝑁 → (1 ≤ (♯‘(1st𝑐)) ↔ 1 ≤ 𝑁))
6462, 63syl5ibrcom 239 . . . . . . . . . . 11 (𝑁 ∈ ℕ → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6564adantl 474 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6665adantr 473 . . . . . . . . 9 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 → 1 ≤ (♯‘(1st𝑐))))
6766pm4.71rd 559 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) ∧ 𝑐 ∈ (ClWalks‘𝐺)) → ((♯‘(1st𝑐)) = 𝑁 ↔ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
6867rabbidva 3370 . . . . . . 7 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
6961, 68syl5eq 2843 . . . . . 6 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7034anbi1i 618 . . . . . . . 8 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ (♯‘(1st𝑐)) = 𝑁) ↔ ((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) = 𝑁))
71 anass 461 . . . . . . . 8 (((𝑐 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝑐))) ∧ (♯‘(1st𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
7270, 71bitri 267 . . . . . . 7 ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∧ (♯‘(1st𝑐)) = 𝑁) ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)))
7372rabbia2 3369 . . . . . 6 {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)}
7469, 43, 733eqtr4g 2856 . . . . 5 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁})
7559, 74reseq12d 5599 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶) = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}))
7645, 75eqtrd 2831 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩)) ↾ {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁}))
77 clwlknf1oclwwlknlem2 27406 . . . . 5 (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7877adantl 474 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})
7978, 43, 733eqtr4g 2856 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐶 = {𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ∣ (♯‘(1st𝑐)) = 𝑁})
80 clwwlkn 27322 . . . 4 (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁}
8180a1i 11 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝑁 ClWWalksN 𝐺) = {𝑠 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑠) = 𝑁})
8276, 79, 81f1oeq123d 6349 . 2 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺) ↔ ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ ((2nd𝑐) substr ⟨0, ((♯‘(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}))
8340, 82mpbird 249 1 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  {crab 3091  cop 4372   class class class wbr 4841  cmpt 4920  cres 5312  1-1-ontowf1o 6098  cfv 6099  (class class class)co 6876  1st c1st 7397  2nd c2nd 7398  0cc0 10222  1c1 10223  cle 10362  cmin 10554  cn 11310  chash 13366   substr csubstr 13661  USPGraphcuspgr 26376  Walkscwlks 26838  ClWalkscclwlks 27016  ClWWalkscclwwlk 27266   ClWWalksN cclwwlkn 27318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-er 7980  df-map 8095  df-pm 8096  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-card 9049  df-cda 9276  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-n0 11577  df-xnn0 11649  df-z 11663  df-uz 11927  df-rp 12071  df-fz 12577  df-fzo 12717  df-hash 13367  df-word 13531  df-lsw 13579  df-concat 13587  df-s1 13612  df-substr 13662  df-pfx 13711  df-edg 26275  df-uhgr 26285  df-upgr 26309  df-uspgr 26378  df-wlks 26841  df-clwlks 27017  df-clwwlk 27267  df-clwwlkn 27320
This theorem is referenced by:  clwlkssizeeqOLD  27412  clwwlknonclwlknonf1oOLD  27723
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