Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkfoclwwlk Structured version   Visualization version   GIF version

Theorem clwlkfoclwwlk 26372
 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 (in an undirected simple graph). (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Hypotheses
Ref Expression
clwlkfclwwlk.1 𝐴 = (1st𝑐)
clwlkfclwwlk.2 𝐵 = (2nd𝑐)
clwlkfclwwlk.c 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
clwlkfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlkfoclwwlk ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→((𝑉 ClWWalksN 𝐸)‘𝑁))
Distinct variable groups:   𝐸,𝑐   𝑁,𝑐   𝑉,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlkfoclwwlk
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.1 . . 3 𝐴 = (1st𝑐)
2 clwlkfclwwlk.2 . . 3 𝐵 = (2nd𝑐)
3 clwlkfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
4 clwlkfclwwlk.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
51, 2, 3, 4clwlkfclwwlk 26371 . 2 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))
6 clwwlknprop 26300 . . . . 5 (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)))
76adantl 481 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)))
8 simpl 472 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁) → 𝑁 ∈ ℕ0)
98anim2i 591 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
10 df-3an 1033 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
119, 10sylibr 223 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
12113adant2 1073 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
1312adantl 481 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
14 isclwwlkn 26297 . . . . . . . 8 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑤) = 𝑁)))
1513, 14syl 17 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑤) = 𝑁)))
16 simpl1 1057 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 𝑉 USGrph 𝐸)
17 simpr2 1061 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 𝑤 ∈ Word 𝑉)
18 eleq1 2676 . . . . . . . . . . . . . . . 16 ((#‘𝑤) = 𝑁 → ((#‘𝑤) ∈ ℙ ↔ 𝑁 ∈ ℙ))
19 prmnn 15226 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) ∈ ℙ → (#‘𝑤) ∈ ℕ)
2019nnge1d 10940 . . . . . . . . . . . . . . . 16 ((#‘𝑤) ∈ ℙ → 1 ≤ (#‘𝑤))
2118, 20syl6bir 243 . . . . . . . . . . . . . . 15 ((#‘𝑤) = 𝑁 → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2221adantl 481 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁) → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
23223ad2ant3 1077 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (𝑁 ∈ ℙ → 1 ≤ (#‘𝑤)))
2423com12 32 . . . . . . . . . . . 12 (𝑁 ∈ ℙ → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤)))
25243ad2ant3 1077 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → 1 ≤ (#‘𝑤)))
2625imp 444 . . . . . . . . . 10 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 1 ≤ (#‘𝑤))
27 clwlkisclwwlk2 26318 . . . . . . . . . 10 ((𝑉 USGrph 𝐸𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (𝑉 ClWWalks 𝐸)))
2816, 17, 26, 27syl3anc 1318 . . . . . . . . 9 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ 𝑤 ∈ (𝑉 ClWWalks 𝐸)))
2928bicomd 212 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ (𝑉 ClWWalks 𝐸) ↔ ∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩)))
3029anbi1d 737 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → ((𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑤) = 𝑁) ↔ (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
3115, 30bitrd 267 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁)))
32 df-br 4584 . . . . . . . . . . 11 (𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ↔ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸))
33 simpl 472 . . . . . . . . . . . . . . 15 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸))
34 prmnn 15226 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
3534nnge1d 10940 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℙ → 1 ≤ 𝑁)
36353ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 1 ≤ 𝑁)
3736adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 1 ≤ 𝑁)
38 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑤) = 𝑁 → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
3938adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
40393ad2ant3 1077 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
4140adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (1 ≤ (#‘𝑤) ↔ 1 ≤ 𝑁))
4237, 41mpbird 246 . . . . . . . . . . . . . . . . . . . 20 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → 1 ≤ (#‘𝑤))
4317, 42jca 553 . . . . . . . . . . . . . . . . . . 19 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
4443adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
45 clwlkswlks 26286 . . . . . . . . . . . . . . . . . 18 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸))
46 wlklenvclwlk 26366 . . . . . . . . . . . . . . . . . 18 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → (#‘𝑓) = (#‘𝑤)))
4744, 45, 46syl2im 39 . . . . . . . . . . . . . . . . 17 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤)))
4847impcom 445 . . . . . . . . . . . . . . . 16 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (#‘𝑓) = (#‘𝑤))
49 vex 3176 . . . . . . . . . . . . . . . . . . . 20 𝑓 ∈ V
50 ovex 6577 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ++ ⟨“(𝑤‘0)”⟩) ∈ V
5149, 50op1st 7067 . . . . . . . . . . . . . . . . . . 19 (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓
5251a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
5352fveq2d 6107 . . . . . . . . . . . . . . . . 17 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
5453ad2antrl 760 . . . . . . . . . . . . . . . 16 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
55 eqcom 2617 . . . . . . . . . . . . . . . . . 18 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5655biimpi 205 . . . . . . . . . . . . . . . . 17 ((#‘𝑤) = 𝑁𝑁 = (#‘𝑤))
5756ad2antll 761 . . . . . . . . . . . . . . . 16 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → 𝑁 = (#‘𝑤))
5848, 54, 573eqtr4d 2654 . . . . . . . . . . . . . . 15 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁)
591fveq2i 6106 . . . . . . . . . . . . . . . . . 18 (#‘𝐴) = (#‘(1st𝑐))
6059eqeq1i 2615 . . . . . . . . . . . . . . . . 17 ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑐)) = 𝑁)
61 fveq2 6103 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st𝑐) = (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
6261fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
6362eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6460, 63syl5bb 271 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6564, 3elrab2 3333 . . . . . . . . . . . . . . 15 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = 𝑁))
6633, 58, 65sylanbrc 695 . . . . . . . . . . . . . 14 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
6743, 45, 46syl2im 39 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤)))
6867ad2antrl 760 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤)))
6968imp 444 . . . . . . . . . . . . . . . . . . . 20 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (#‘𝑓) = (#‘𝑤))
7069opeq2d 4347 . . . . . . . . . . . . . . . . . . 19 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ⟨0, (#‘𝑓)⟩ = ⟨0, (#‘𝑤)⟩)
7170oveq2d 6565 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
72 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸))
73 eqeq2 2621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 = (#‘𝑤) → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
7473eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑤) = 𝑁 → ((#‘𝑓) = 𝑁 ↔ (#‘𝑓) = (#‘𝑤)))
7574imbi2d 329 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑤) = 𝑁 → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤))))
7675ad2antll 761 . . . . . . . . . . . . . . . . . . . . . 22 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = 𝑁) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = (#‘𝑤))))
7768, 76mpbird 246 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → (#‘𝑓) = 𝑁))
7877imp 444 . . . . . . . . . . . . . . . . . . . 20 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (#‘𝑓) = 𝑁)
7951a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = 𝑓)
8079fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓))
8162, 80eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (#‘(1st𝑐)) = (#‘𝑓))
8281eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘(1st𝑐)) = 𝑁 ↔ (#‘𝑓) = 𝑁))
8360, 82syl5bb 271 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((#‘𝐴) = 𝑁 ↔ (#‘𝑓) = 𝑁))
8483, 3elrab2 3333 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (#‘𝑓) = 𝑁))
8572, 78, 84sylanbrc 695 . . . . . . . . . . . . . . . . . . 19 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶)
86 ovex 6577 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V
8759opeq2i 4344 . . . . . . . . . . . . . . . . . . . . . 22 ⟨0, (#‘𝐴)⟩ = ⟨0, (#‘(1st𝑐))⟩
882, 87oveq12i 6561 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩)
89 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (2nd𝑐) = (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
9062opeq2d 4347 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ⟨0, (#‘(1st𝑐))⟩ = ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩)
9189, 90oveq12d 6567 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩))
9249, 50op2nd 7068 . . . . . . . . . . . . . . . . . . . . . . 23 (2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = (𝑤 ++ ⟨“(𝑤‘0)”⟩)
9351fveq2i 6106 . . . . . . . . . . . . . . . . . . . . . . . 24 (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)) = (#‘𝑓)
9493opeq2i 4344 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩ = ⟨0, (#‘𝑓)⟩
9592, 94oveq12i 6561 . . . . . . . . . . . . . . . . . . . . . 22 ((2nd ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) substr ⟨0, (#‘(1st ‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩)
9691, 95syl6eq 2660 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((2nd𝑐) substr ⟨0, (#‘(1st𝑐))⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9788, 96syl5eq 2656 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐵 substr ⟨0, (#‘𝐴)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9897, 4fvmptg 6189 . . . . . . . . . . . . . . . . . . 19 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ 𝐶 ∧ ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩) ∈ V) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
9985, 86, 98sylancl 693 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑓)⟩))
10043ad2antrl 760 . . . . . . . . . . . . . . . . . . . 20 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
101100adantr 480 . . . . . . . . . . . . . . . . . . 19 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → (𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)))
102 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → 𝑤 ∈ Word 𝑉)
103 wrdsymb1 13197 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (𝑤‘0) ∈ 𝑉)
104103s1cld 13236 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → ⟨“(𝑤‘0)”⟩ ∈ Word 𝑉)
105 eqidd 2611 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → (#‘𝑤) = (#‘𝑤))
106 swrdccatid 13348 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤 ∈ Word 𝑉 ∧ ⟨“(𝑤‘0)”⟩ ∈ Word 𝑉 ∧ (#‘𝑤) = (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
107102, 104, 105, 106syl3anc 1318 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩) = 𝑤)
108107eqcomd 2616 . . . . . . . . . . . . . . . . . . 19 ((𝑤 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑤)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
109101, 108syl 17 . . . . . . . . . . . . . . . . . 18 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → 𝑤 = ((𝑤 ++ ⟨“(𝑤‘0)”⟩) substr ⟨0, (#‘𝑤)⟩))
11071, 99, 1093eqtr4rd 2655 . . . . . . . . . . . . . . . . 17 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸)) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
111110ex 449 . . . . . . . . . . . . . . . 16 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
112111adantr 480 . . . . . . . . . . . . . . 15 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
113 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝐹𝑐) = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))
114113eqeq2d 2620 . . . . . . . . . . . . . . . . 17 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → (𝑤 = (𝐹𝑐) ↔ 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩)))
115114imbi2d 329 . . . . . . . . . . . . . . . 16 (𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
116115adantl 481 . . . . . . . . . . . . . . 15 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹𝑐)) ↔ (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹‘⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩))))
117112, 116mpbird 246 . . . . . . . . . . . . . 14 (((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) ∧ 𝑐 = ⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → 𝑤 = (𝐹𝑐)))
11866, 117rspcimedv 3284 . . . . . . . . . . . . 13 ((⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) ∧ (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁)) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
119118ex 449 . . . . . . . . . . . 12 (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
120119pm2.43b 53 . . . . . . . . . . 11 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (⟨𝑓, (𝑤 ++ ⟨“(𝑤‘0)”⟩)⟩ ∈ (𝑉 ClWalks 𝐸) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
12132, 120syl5bi 231 . . . . . . . . . 10 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
122121exlimdv 1848 . . . . . . . . 9 ((((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) ∧ (#‘𝑤) = 𝑁) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
123122ex 449 . . . . . . . 8 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → ((#‘𝑤) = 𝑁 → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
124123com23 84 . . . . . . 7 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) → ((#‘𝑤) = 𝑁 → ∃𝑐𝐶 𝑤 = (𝐹𝑐))))
125124impd 446 . . . . . 6 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → ((∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑤 ++ ⟨“(𝑤‘0)”⟩) ∧ (#‘𝑤) = 𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
12631, 125sylbid 229 . . . . 5 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁))) → (𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
127126impancom 455 . . . 4 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑤 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑤) = 𝑁)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1287, 127mpd 15 . . 3 (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ∃𝑐𝐶 𝑤 = (𝐹𝑐))
129128ralrimiva 2949 . 2 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → ∀𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)∃𝑐𝐶 𝑤 = (𝐹𝑐))
130 dffo3 6282 . 2 (𝐹:𝐶onto→((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ ∀𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)∃𝑐𝐶 𝑤 = (𝐹𝑐)))
1315, 129, 130sylanbrc 695 1 ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→((𝑉 ClWWalksN 𝐸)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = 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  Fincfn 7841  0cc0 9815  1c1 9816   ≤ cle 9954  ℕ0cn0 11169  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  ℙcprime 15223   USGrph cusg 25859   Walks cwalk 26026   ClWalks cclwlk 26275   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277 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-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-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-usgra 25862  df-wlk 26036  df-clwlk 26278  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  clwlkf1oclwwlk  26378
 Copyright terms: Public domain W3C validator