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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-onto→wf1o 6098  ‘cfv 6099  (class class class)co 6876  1st c1st 7397  2nd c2nd 7398  0cc0 10222  1c1 10223   ≤ cle 10362   − cmin 10554  ℕcn 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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