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Theorem clwlksfoclwwlk 26836
 Description: There is an onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.)
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
clwlksfoclwwlk ((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 clwlksfoclwwlk
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, 4clwlksfclwwlk 26835 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))
6 eqid 2621 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
76clwwlknbp 26759 . . . . 5 (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))
87adantl 482 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))
9 prmnn 15315 . . . . . . . . 9 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
109ad2antlr 762 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑁 ∈ ℕ)
11 isclwwlksn 26756 . . . . . . . 8 (𝑁 ∈ ℕ → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑤 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑤) = 𝑁)))
1210, 11syl 17 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑤 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑤) = 𝑁)))
13 fusgrusgr 26109 . . . . . . . . . . . . 13 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
14 usgruspgr 25973 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
1513, 14syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph )
1615adantr 481 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USPGraph )
1716adantr 481 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝐺 ∈ USPGraph )
18 simprl 793 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑤 ∈ Word (Vtx‘𝐺))
19 eleq1 2686 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 𝑁 → ((#‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
20 prmnn 15315 . . . . . . . . . . . . . . . 16 ((#‘𝑤) ∈ ℙ → (#‘𝑤) ∈ ℕ)
2120nnge1d 11010 . . . . . . . . . . . . . . 15 ((#‘𝑤) ∈ ℙ → 1 ≤ (#‘𝑤))
2219, 21syl6bir 244 . . . . . . . . . . . . . 14 ((#‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2322adantl 482 . . . . . . . . . . . . 13 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2423com12 32 . . . . . . . . . . . 12 (𝑁 ∈ ℙ → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → 1 ≤ (#‘𝑤)))
2524adantl 482 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → 1 ≤ (#‘𝑤)))
2625imp 445 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤))
27 eqid 2621 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
286, 27clwlkclwwlk2 26778 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
2917, 18, 26, 28syl3anc 1323 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalks‘𝐺)))
3029bicomd 213 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (ClWWalks‘𝐺) ↔ ∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
3130anbi1d 740 . . . . . . 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)))
3212, 31bitrd 268 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) ↔ (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
33 df-br 4616 . . . . . . . . 9 (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
34 simpl 473 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
359nnge1d 11010 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℙ → 1 ≤ 𝑁)
3635ad2antlr 762 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ 𝑁)
37 breq2 4619 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑤) = 𝑁 → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3837ad2antll 764 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3936, 38mpbird 247 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤))
4018, 39jca 554 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)))
41 clwlkwlk 26547 . . . . . . . . . . . . . . . 16 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺))
42 wlklenvclwlk 26427 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (Walks‘𝐺) → (#‘𝑓) = (#‘𝑤)))
4340, 41, 42syl2im 40 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤)))
4443impcom 446 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘𝑓) = (#‘𝑤))
45 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
46 ovex 6635 . . . . . . . . . . . . . . . . . 18 (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
4745, 46op1st 7124 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓
4847a1i 11 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
4948fveq2d 6154 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
5049adantl 482 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
51 eqcom 2628 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5251biimpi 206 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5352ad2antll 764 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑁 = (#‘𝑤))
5453adantl 482 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → 𝑁 = (#‘𝑤))
5544, 50, 543eqtr4d 2665 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
561fveq2i 6153 . . . . . . . . . . . . . . . 16 (#‘𝐴) = (#‘(1st𝑐))
5756eqeq1i 2626 . . . . . . . . . . . . . . 15 ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑐)) = 𝑁)
58 fveq2 6150 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
5958fveq2d 6154 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
6059eqeq1d 2623 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6157, 60syl5bb 272 . . . . . . . . . . . . . 14 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6261, 3elrab2 3349 . . . . . . . . . . . . 13 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6334, 55, 62sylanbrc 697 . . . . . . . . . . . 12 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
6444adantr 481 . . . . . . . . . . . . . . . . . 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))
6564opeq2d 4379 . . . . . . . . . . . . . . . . 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)⟩)
6665oveq2d 6623 . . . . . . . . . . . . . . . 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)⟩))
67 simpr 477 . . . . . . . . . . . . . . . . . 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))
6843adantl 482 . . . . . . . . . . . . . . . . . . . 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)))
69 eqeq2 2632 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 = (#‘𝑤) → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
7069eqcoms 2629 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑤) = 𝑁 → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
7170imbi2d 330 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (#‘𝑓) = (#‘𝑤))))
7271ad2antll 764 . . . . . . . . . . . . . . . . . . . . 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))))
7372adantl 482 . . . . . . . . . . . . . . . . . . . 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))))
7468, 73mpbird 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))
7574imp 445 . . . . . . . . . . . . . . . . . 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)
7647a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
7776fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
7859, 77eqtrd 2655 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘𝑓))
7978eqeq1d 2623 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘𝑓) = 𝑁))
8057, 79syl5bb 272 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘𝑓) = 𝑁))
8180, 3elrab2 3349 . . . . . . . . . . . . . . . . . 18 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ (#‘𝑓) = 𝑁))
8267, 75, 81sylanbrc 697 . . . . . . . . . . . . . . . . 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)
83 ovex 6635 . . . . . . . . . . . . . . . . 17 ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V
8456opeq2i 4376 . . . . . . . . . . . . . . . . . . . 20 ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st𝑐))⟩
852, 84oveq12i 6619 . . . . . . . . . . . . . . . . . . 19 (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩)
86 fveq2 6150 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
8759opeq2d 4379 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (#‘(1st𝑐))⟩ = ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
8886, 87oveq12d 6625 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
8945, 46op2nd 7125 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
9047fveq2i 6153 . . . . . . . . . . . . . . . . . . . . . 22 (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓)
9190opeq2i 4376 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (#‘𝑓)⟩
9289, 91oveq12i 6619 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩)
9388, 92syl6eq 2671 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9485, 93syl5eq 2667 . . . . . . . . . . . . . . . . . 18 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9594, 4fvmptg 6239 . . . . . . . . . . . . . . . . 17 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9682, 83, 95sylancl 693 . . . . . . . . . . . . . . . 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)⟩))
9740ad2antlr 762 . . . . . . . . . . . . . . . . 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)))
98 simpl 473 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → 𝑤 ∈ Word (Vtx‘𝐺))
99 wrdsymb1 13284 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
10099s1cld 13325 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
101 eqidd 2622 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (#‘𝑤) = (#‘𝑤))
102 swrdccatid 13437 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
10398, 100, 101, 102syl3anc 1323 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
104103eqcomd 2627 . . . . . . . . . . . . . . . . 17 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
10597, 104syl 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)⟩))
10666, 96, 1053eqtr4rd 2666 . . . . . . . . . . . . . . 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)”⟩)⟩))
107106ex 450 . . . . . . . . . . . . . 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)”⟩)⟩)))
108107adantr 481 . . . . . . . . . . . . 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)”⟩)⟩)))
109 fveq2 6150 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
110109eqeq2d 2631 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
111110imbi2d 330 . . . . . . . . . . . . . 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)”⟩)⟩))))
112111adantl 482 . . . . . . . . . . . . 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)”⟩)⟩))))
113108, 112mpbird 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)))
11463, 113rspcimedv 3297 . . . . . . . . . . 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)))
115114ex 450 . . . . . . . . . 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))))
116115pm2.43b 55 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
11733, 116syl5bi 232 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
118117exlimdv 1858 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
119118adantrd 484 . . . . . 6 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → ((∃𝑓 𝑓(ClWalks‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
12032, 119sylbid 230 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalksN 𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
121120impancom 456 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1228, 121mpd 15 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))
123122ralrimiva 2960 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐))
124 dffo3 6332 . 2 (𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺) ∧ ∀𝑤 ∈ (𝑁 ClWWalksN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1255, 123, 124sylanbrc 697 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911  Vcvv 3186  ⟨cop 4156   class class class wbr 4615   ↦ cmpt 4675  ⟶wf 5845  –onto→wfo 5847  ‘cfv 5849  (class class class)co 6607  1st c1st 7114  2nd c2nd 7115  0cc0 9883  1c1 9884   ≤ cle 10022  ℕcn 10967  #chash 13060  Word cword 13233   ++ cconcat 13235  ⟨“cs1 13236   substr csubstr 13237  ℙcprime 15312  Vtxcvtx 25781  iEdgciedg 25782   USPGraph cuspgr 25943   USGraph cusgr 25944   FinUSGraph cfusgr 26103  Walkscwlks 26369  ClWalkscclwlks 26542  ClWWalkscclwwlks 26749   ClWWalksN cclwwlksn 26750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-sup 8295  df-card 8712  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-3 11027  df-n0 11240  df-xnn0 11311  df-z 11325  df-uz 11635  df-rp 11780  df-fz 12272  df-fzo 12410  df-seq 12745  df-exp 12804  df-hash 13061  df-word 13241  df-lsw 13242  df-concat 13243  df-s1 13244  df-substr 13245  df-cj 13776  df-re 13777  df-im 13778  df-sqrt 13912  df-abs 13913  df-dvds 14911  df-prm 15313  df-edg 25847  df-uhgr 25856  df-upgr 25880  df-uspgr 25945  df-usgr 25946  df-fusgr 26104  df-wlks 26372  df-clwlks 26543  df-clwwlks 26751  df-clwwlksn 26752 This theorem is referenced by:  clwlksf1oclwwlk  26843
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