Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwlksfoclwwlk Structured version   Visualization version   GIF version

Theorem clwlksfoclwwlk 41270
 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 41269 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺))
6 eqid 2610 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
76clwwlknbp 41193 . . . . 5 (𝑤 ∈ (𝑁 ClWWalkSN 𝐺) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))
87adantl 481 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ (𝑁 ClWWalkSN 𝐺)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))
9 prmnn 15226 . . . . . . . . 9 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
109ad2antlr 759 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑁 ∈ ℕ)
11 isclwwlksn 41190 . . . . . . . 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 40541 . . . . . . . . . . . . 13 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
14 usgruspgr 40408 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph )
1513, 14syl 17 . . . . . . . . . . . 12 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph )
1615adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐺 ∈ USPGraph )
1716adantr 480 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝐺 ∈ USPGraph )
18 simprl 790 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑤 ∈ Word (Vtx‘𝐺))
19 eleq1 2676 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 𝑁 → ((#‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
20 prmnn 15226 . . . . . . . . . . . . . . . 16 ((#‘𝑤) ∈ ℙ → (#‘𝑤) ∈ ℕ)
2120nnge1d 10940 . . . . . . . . . . . . . . 15 ((#‘𝑤) ∈ ℙ → 1 ≤ (#‘𝑤))
2219, 21syl6bir 243 . . . . . . . . . . . . . 14 ((#‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2322adantl 481 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ((𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁) → 1 ≤ (#‘𝑤)))
2625imp 444 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤))
27 eqid 2610 . . . . . . . . . . 11 (iEdg‘𝐺) = (iEdg‘𝐺)
286, 27clwlkclwwlk2 41212 . . . . . . . . . 10 ((𝐺 ∈ USPGraph ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (∃𝑓 𝑓(ClWalkS‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalkS‘𝐺)))
2917, 18, 26, 28syl3anc 1318 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalkS‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (ClWWalkS‘𝐺)))
3029bicomd 212 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (ClWWalkS‘𝐺) ↔ ∃𝑓 𝑓(ClWalkS‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
3130anbi1d 737 . . . . . . 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 267 . . . . . 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 4584 . . . . . . . . 9 (𝑓(ClWalkS‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺))
34 simpl 472 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺))
359nnge1d 10940 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ ℙ → 1 ≤ 𝑁)
3635ad2antlr 759 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ 𝑁)
37 breq2 4587 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑤) = 𝑁 → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3837ad2antll 761 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3936, 38mpbird 246 . . . . . . . . . . . . . . . . 17 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤))
4018, 39jca 553 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)))
41 clwlk1wlk 40982 . . . . . . . . . . . . . . . 16 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (1Walks‘𝐺))
42 1wlklenvclwlk 40863 . . . . . . . . . . . . . . . 16 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (1Walks‘𝐺) → (#‘𝑓) = (#‘𝑤)))
4340, 41, 42syl2im 39 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) → (#‘𝑓) = (#‘𝑤)))
4443impcom 445 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘𝑓) = (#‘𝑤))
45 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑓 ∈ V
46 ovex 6577 . . . . . . . . . . . . . . . . . 18 (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
4745, 46op1st 7067 . . . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
5049adantl 481 . . . . . . . . . . . . . 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 2617 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5251biimpi 205 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5352ad2antll 761 . . . . . . . . . . . . . . 15 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → 𝑁 = (#‘𝑤))
5453adantl 481 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → 𝑁 = (#‘𝑤))
5544, 50, 543eqtr4d 2654 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
561fveq2i 6106 . . . . . . . . . . . . . . . 16 (#‘𝐴) = (#‘(1st𝑐))
5756eqeq1i 2615 . . . . . . . . . . . . . . 15 ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑐)) = 𝑁)
58 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
5958fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
6059eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6157, 60syl5bb 271 . . . . . . . . . . . . . 14 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6261, 3elrab2 3333 . . . . . . . . . . . . 13 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6334, 55, 62sylanbrc 695 . . . . . . . . . . . 12 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁))) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
6444adantr 480 . . . . . . . . . . . . . . . . . 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 4347 . . . . . . . . . . . . . . . . 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 6565 . . . . . . . . . . . . . . . 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 476 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 = (#‘𝑤) → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
7069eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑤) = 𝑁 → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
7170imbi2d 329 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) → (#‘𝑓) = (#‘𝑤))))
7271ad2antll 761 . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
7859, 77eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘𝑓))
7978eqeq1d 2612 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘𝑓) = 𝑁))
8057, 79syl5bb 271 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘𝑓) = 𝑁))
8180, 3elrab2 3333 . . . . . . . . . . . . . . . . . 18 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) ∧ (#‘𝑓) = 𝑁))
8267, 75, 81sylanbrc 695 . . . . . . . . . . . . . . . . 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 6577 . . . . . . . . . . . . . . . . 17 ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V
8456opeq2i 4344 . . . . . . . . . . . . . . . . . . . 20 ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st𝑐))⟩
852, 84oveq12i 6561 . . . . . . . . . . . . . . . . . . 19 (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩)
86 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
8759opeq2d 4347 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (#‘(1st𝑐))⟩ = ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
8886, 87oveq12d 6567 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
8945, 46op2nd 7068 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
9047fveq2i 6106 . . . . . . . . . . . . . . . . . . . . . 22 (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓)
9190opeq2i 4344 . . . . . . . . . . . . . . . . . . . . 21 ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (#‘𝑓)⟩
9289, 91oveq12i 6561 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩)
9388, 92syl6eq 2660 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9485, 93syl5eq 2656 . . . . . . . . . . . . . . . . . 18 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9594, 4fvmptg 6189 . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → 𝑤 ∈ Word (Vtx‘𝐺))
99 wrdsymb1 13197 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (𝑤‘0) ∈ (Vtx‘𝐺))
10099s1cld 13236 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺))
101 eqidd 2611 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → (#‘𝑤) = (#‘𝑤))
102 swrdccatid 13348 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ ⟨“(𝑤‘0)”⟩ ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
10398, 100, 101, 102syl3anc 1318 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
104103eqcomd 2616 . . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
110109eqeq2d 2620 . . . . . . . . . . . . . . 15 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
111110imbi2d 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)”⟩)⟩))))
112111adantl 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 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
113108, 112mpbird 246 . . . . . . . . . . . 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 3284 . . . . . . . . . . 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 449 . . . . . . . . . 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 53 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (ClWalkS‘𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
11733, 116syl5bi 231 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑓(ClWalkS‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
118117exlimdv 1848 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (∃𝑓 𝑓(ClWalkS‘𝐺)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
119118adantrd 483 . . . . . 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 229 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) ∧ (𝑤 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ (𝑁 ClWWalkSN 𝐺) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
121120impancom 455 . . . 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 2949 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ (𝑁 ClWWalkSN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐))
124 dffo3 6282 . 2 (𝐹:𝐶onto→(𝑁 ClWWalkSN 𝐺) ↔ (𝐹:𝐶⟶(𝑁 ClWWalkSN 𝐺) ∧ ∀𝑤 ∈ (𝑁 ClWWalkSN 𝐺)∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1255, 123, 124sylanbrc 695 1 ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalkSN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   ≤ cle 9954  ℕcn 10897  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  ℙcprime 15223  Vtxcvtx 25673  iEdgciedg 25674   USPGraph cuspgr 40378   USGraph cusgr 40379   FinUSGraph cfusgr 40535  1Walksc1wlks 40796  ClWalkScclwlks 40976  ClWWalkScclwwlks 41183   ClWWalkSN cclwwlksn 41184 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-prm 15224  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-fusgr 40536  df-1wlks 40800  df-wlks 40801  df-clwlks 40977  df-clwwlks 41185  df-clwwlksn 41186 This theorem is referenced by:  clwlksf1oclwwlk  41277
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