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Theorem clwlksfoclwwlkOLD 27244
Description: Obsolete version of clwlkclwwlkfo 27159 as of 25-May-2022. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
Assertion
Ref Expression
clwlksfoclwwlkOLD ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksfoclwwlkOLD
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlksfclwwlk.1 . . 3 𝐴 = (1st𝑐)
2 clwlksfclwwlk.2 . . 3 𝐵 = (2nd𝑐)
3 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4 clwlksfclwwlk.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
51, 2, 3, 4clwlksfclwwlkOLD 27243 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))
6 eqid 2771 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
76clwwlknbp 27190 . . . . 5 (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))
87adantl 467 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))
9 isclwwlkn 27180 . . . . . . 7 (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑤 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑤) = 𝑁))
10 fusgrusgr 26437 . . . . . . . . . . . . 13 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
11 usgruspgr 26295 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
1210, 11syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph)
1312adantr 466 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USPGraph)
1413adantr 466 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 𝐺 ∈ USPGraph)
15 simprl 754 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 𝑤 ∈ Word (Vtx‘𝐺))
16 eleq1 2838 . . . . . . . . . . . . . . 15 ((♯‘𝑤) = 𝑁 → ((♯‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
17 prmnn 15595 . . . . . . . . . . . . . . . 16 ((♯‘𝑤) ∈ ℙ → (♯‘𝑤) ∈ ℕ)
1817nnge1d 11269 . . . . . . . . . . . . . . 15 ((♯‘𝑤) ∈ ℙ → 1 ≤ (♯‘𝑤))
1916, 18syl6bir 244 . . . . . . . . . . . . . 14 ((♯‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (♯‘𝑤)))
2019adantl 467 . . . . . . . . . . . . 13 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → (𝑁 ∈ ℙ → 1 ≤ (♯‘𝑤)))
2120com12 32 . . . . . . . . . . . 12 (𝑁 ∈ ℙ → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → 1 ≤ (♯‘𝑤)))
2221adantl 467 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → 1 ≤ (♯‘𝑤)))
2322imp 393 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 1 ≤ (♯‘𝑤))
24 eqid 2771 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
256, 24clwlkclwwlk2 27153 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
2614, 15, 23, 25syl3anc 1476 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
2726bicomd 213 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ ∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
2827anbi1d 615 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → ((𝑤 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑤) = 𝑁) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (♯‘𝑤) = 𝑁)))
299, 28syl5bb 272 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (♯‘𝑤) = 𝑁)))
30 df-br 4788 . . . . . . . . 9 (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
31 simpl 468 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
32 prmnn 15595 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
3332nnge1d 11269 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℙ → 1 ≤ 𝑁)
3433ad2antlr 706 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 1 ≤ 𝑁)
35 breq2 4791 . . . . . . . . . . . . . . . . . . 19 ((♯‘𝑤) = 𝑁 → (1 ≤ (♯‘𝑤) ↔ 1 ≤ 𝑁))
3635ad2antll 708 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (1 ≤ (♯‘𝑤) ↔ 1 ≤ 𝑁))
3734, 36mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 1 ≤ (♯‘𝑤))
3815, 37jca 501 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
39 clwlkwlk 26906 . . . . . . . . . . . . . . . 16 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺))
40 wlklenvclwlk 26786 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝑓) = (♯‘𝑤)))
4138, 39, 40syl2im 40 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤)))
4241impcom 394 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (♯‘𝑓) = (♯‘𝑤))
43 vex 3354 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
44 ovex 6827 . . . . . . . . . . . . . . . . . 18 (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
4543, 44op1st 7327 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓
4645a1i 11 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
4746fveq2d 6337 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓))
4847adantl 467 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓))
49 eqcom 2778 . . . . . . . . . . . . . . . . 17 ((♯‘𝑤) = 𝑁𝑁 = (♯‘𝑤))
5049biimpi 206 . . . . . . . . . . . . . . . 16 ((♯‘𝑤) = 𝑁𝑁 = (♯‘𝑤))
5150ad2antll 708 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → 𝑁 = (♯‘𝑤))
5251adantl 467 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → 𝑁 = (♯‘𝑤))
5342, 48, 523eqtr4d 2815 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
541fveq2i 6336 . . . . . . . . . . . . . . . 16 (♯‘𝐴) = (♯‘(1st𝑐))
5554eqeq1i 2776 . . . . . . . . . . . . . . 15 ((♯‘𝐴) = 𝑁 ↔ (♯‘(1st𝑐)) = 𝑁)
56 fveq2 6333 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
5756fveq2d 6337 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(1st𝑐)) = (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
5857eqeq1d 2773 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(1st𝑐)) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
5955, 58syl5bb 272 . . . . . . . . . . . . . 14 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘𝐴) = 𝑁 ↔ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6059, 3elrab2 3518 . . . . . . . . . . . . 13 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6131, 53, 60sylanbrc 572 . . . . . . . . . . . 12 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
6242adantr 466 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = (♯‘𝑤))
6362opeq2d 4547 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨0, (♯‘𝑓)⟩ = ⟨0, (♯‘𝑤)⟩)
6463oveq2d 6812 . . . . . . . . . . . . . . . 16 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
65 simpr 471 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
6641adantl 467 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤)))
67 eqeq2 2782 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 = (♯‘𝑤) → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑓) = (♯‘𝑤)))
6867eqcoms 2779 . . . . . . . . . . . . . . . . . . . . . . 23 ((♯‘𝑤) = 𝑁 → ((♯‘𝑓) = 𝑁 ↔ (♯‘𝑓) = (♯‘𝑤)))
6968imbi2d 329 . . . . . . . . . . . . . . . . . . . . . 22 ((♯‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤))))
7069ad2antll 708 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤))))
7170adantl 467 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = (♯‘𝑤))))
7266, 71mpbird 247 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = 𝑁))
7372imp 393 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = 𝑁)
7445a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
7574fveq2d 6337 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓))
7657, 75eqtrd 2805 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (♯‘(1st𝑐)) = (♯‘𝑓))
7776eqeq1d 2773 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘(1st𝑐)) = 𝑁 ↔ (♯‘𝑓) = 𝑁))
7855, 77syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((♯‘𝐴) = 𝑁 ↔ (♯‘𝑓) = 𝑁))
7978, 3elrab2 3518 . . . . . . . . . . . . . . . . . 18 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (♯‘𝑓) = 𝑁))
8065, 73, 79sylanbrc 572 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
81 ovex 6827 . . . . . . . . . . . . . . . . 17 ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩) ∈ V
8254opeq2i 4544 . . . . . . . . . . . . . . . . . . . 20 ⟨0, (♯‘𝐴)⟩ = ⟨0, (♯‘(1st𝑐))⟩
832, 82oveq12i 6808 . . . . . . . . . . . . . . . . . . 19 (𝐵 substr ⟨0, (♯‘𝐴)⟩) = ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩)
84 fveq2 6333 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
8557opeq2d 4547 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (♯‘(1st𝑐))⟩ = ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
8684, 85oveq12d 6814 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
8743, 44op2nd 7328 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
8845fveq2i 6336 . . . . . . . . . . . . . . . . . . . . . 22 (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (♯‘𝑓)
8988opeq2i 4544 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (♯‘𝑓)⟩
9087, 89oveq12i 6808 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (♯‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩)
9186, 90syl6eq 2821 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (♯‘(1st𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
9283, 91syl5eq 2817 . . . . . . . . . . . . . . . . . 18 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (♯‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
9392, 4fvmptg 6424 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
9480, 81, 93sylancl 574 . . . . . . . . . . . . . . . 16 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑓)⟩))
9538ad2antlr 706 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)))
96 simpl 468 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → 𝑤 ∈ Word (Vtx‘𝐺))
97 wrdsymb1 13539 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
9897s1cld 13583 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
99 eqidd 2772 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → (♯‘𝑤) = (♯‘𝑤))
100 swrdccatid 13706 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩) = 𝑤)
10196, 98, 99, 100syl3anc 1476 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩) = 𝑤)
102101eqcomd 2777 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑤)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
10395, 102syl 17 . . . . . . . . . . . . . . . 16 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (♯‘𝑤)⟩))
10464, 94, 1033eqtr4rd 2816 . . . . . . . . . . . . . . 15 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
105104ex 397 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
106105adantr 466 . . . . . . . . . . . . 13 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
107 fveq2 6333 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
108107eqeq2d 2781 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
109108imbi2d 329 . . . . . . . . . . . . . 14 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
110109adantl 467 . . . . . . . . . . . . 13 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
111106, 110mpbird 247 . . . . . . . . . . . 12 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → 𝑤 = (𝐹𝑐)))
11261, 111rspcimedv 3462 . . . . . . . . . . 11 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
113112ex 397 . . . . . . . . . 10 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
114113pm2.43b 55 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
11530, 114syl5bi 232 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
116115exlimdv 2013 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
117116adantrd 479 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → ((∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (♯‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
11829, 117sylbid 230 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
119118impancom 439 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1208, 119mpd 15 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))
121120ralrimiva 3115 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐))
122 dffo3 6519 . 2 (𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1235, 121, 122sylanbrc 572 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  {crab 3065  Vcvv 3351  cop 4323   class class class wbr 4787  cmpt 4864  wf 6026  ontowfo 6028  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  0cc0 10142  1c1 10143  cle 10281  chash 13321  Word cword 13487   ++ cconcat 13489  ⟨“cs1 13490   substr csubstr 13491  cprime 15592  Vtxcvtx 26095  iEdgciedg 26096  USPGraphcuspgr 26265  USGraphcusgr 26266  FinUSGraphcfusgr 26431  Walkscwlks 26727  ClWalkscclwlks 26901  ClWWalkscclwwlk 27131   ClWWalksN cclwwlkn 27174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-div 10891  df-nn 11227  df-2 11285  df-3 11286  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-rp 12036  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-dvds 15190  df-prm 15593  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-uspgr 26267  df-usgr 26268  df-fusgr 26432  df-wlks 26730  df-clwlks 26902  df-clwwlk 27132  df-clwwlkn 27176
This theorem is referenced by:  clwlksf1oclwwlkOLD  27251
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