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Theorem 3v3e3cycl1 26172
 Description: If there is a cycle of length 3 in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Assertion
Ref Expression
3v3e3cycl1 ((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑐   𝑃,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐
Allowed substitution hints:   𝐹(𝑎,𝑏,𝑐)

Proof of Theorem 3v3e3cycl1
Dummy variables 𝑘 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cycliswlk 26160 . . . . 5 (𝐹(𝑉 Cycles 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)
2 wlkbprop 26051 . . . . 5 (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
31, 2syl 17 . . . 4 (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
4 iscycl 26153 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ↔ (𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
5 ispth 26098 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))
6 istrl 26067 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
7 fzo0to3tp 12421 . . . . . . . . . . . . . . . . . . 19 (0..^3) = {0, 1, 2}
87raleqi 3119 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
9 0z 11265 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℤ
10 1z 11284 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
11 2z 11286 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
12 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → (𝐹𝑘) = (𝐹‘0))
1312fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘0)))
14 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → (𝑃𝑘) = (𝑃‘0))
15 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
16 0p1e1 11009 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 + 1) = 1
1715, 16syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → (𝑘 + 1) = 1)
1817fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1))
1914, 18preq12d 4220 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 0 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)})
2013, 19eqeq12d 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 0 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
21 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 1 → (𝐹𝑘) = (𝐹‘1))
2221fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 1 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘1)))
23 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 1 → (𝑃𝑘) = (𝑃‘1))
24 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 1 → (𝑘 + 1) = (1 + 1))
25 1p1e2 11011 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 + 1) = 2
2624, 25syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 1 → (𝑘 + 1) = 2)
2726fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2))
2823, 27preq12d 4220 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 1 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)})
2922, 28eqeq12d 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 1 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
30 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 2 → (𝐹𝑘) = (𝐹‘2))
3130fveq2d 6107 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 2 → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘2)))
32 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 2 → (𝑃𝑘) = (𝑃‘2))
33 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
34 2p1e3 11028 . . . . . . . . . . . . . . . . . . . . . . . 24 (2 + 1) = 3
3533, 34syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 2 → (𝑘 + 1) = 3)
3635fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
3732, 36preq12d 4220 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 2 → {(𝑃𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
3831, 37eqeq12d 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 2 → ((𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
3920, 29, 38raltpg 4183 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) → (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
409, 10, 11, 39mp3an 1416 . . . . . . . . . . . . . . . . . 18 (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
418, 40bitri 263 . . . . . . . . . . . . . . . . 17 (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
42 preq2 4213 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃‘3) = (𝑃‘0) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
4342eqcoms 2618 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
4443eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃‘0) = (𝑃‘3) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}))
45443anbi3d 1397 . . . . . . . . . . . . . . . . . . . 20 ((𝑃‘0) = (𝑃‘3) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}) ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)})))
46 3pos 10991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 0 < 3
47 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 3 → (0 < (#‘𝐹) ↔ 0 < 3))
4846, 47mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 3 → 0 < (#‘𝐹))
49 0nn0 11184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 0 ∈ ℕ0
5048, 49jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 3 → (0 ∈ ℕ0 ∧ 0 < (#‘𝐹)))
51 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (0 ∈ ℕ0 ∧ 0 < (#‘𝐹))) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
5250, 51sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 3) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
53 1lt3 11073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1 < 3
54 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 3 → (1 < (#‘𝐹) ↔ 1 < 3))
5553, 54mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 3 → 1 < (#‘𝐹))
56 1nn0 11185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 ∈ ℕ0
5755, 56jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 3 → (1 ∈ ℕ0 ∧ 1 < (#‘𝐹)))
58 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (1 ∈ ℕ0 ∧ 1 < (#‘𝐹))) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
5957, 58sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 3) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
60 2lt3 11072 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 < 3
61 breq2 4587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((#‘𝐹) = 3 → (2 < (#‘𝐹) ↔ 2 < 3))
6260, 61mpbiri 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((#‘𝐹) = 3 → 2 < (#‘𝐹))
63 2nn0 11186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℕ0
6462, 63jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝐹) = 3 → (2 ∈ ℕ0 ∧ 2 < (#‘𝐹)))
65 nvnencycllem 26171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (2 ∈ ℕ0 ∧ 2 < (#‘𝐹))) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)} → {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸))
6664, 65sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 3) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)} → {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸))
6752, 59, 663anim123d 1398 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Fun 𝐸𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 3) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸)))
6867adantlrr 753 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Fun 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹)) ∧ (#‘𝐹) = 3) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸)))
6968imp 444 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((Fun 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹)) ∧ (#‘𝐹) = 3) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)})) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸))
70 3z 11287 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 ∈ ℤ
71 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (3 ∈ ℤ → 3 ∈ (ℤ‘3))
7270, 71ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 ∈ (ℤ‘3)
73 4fvwrd4 12328 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((3 ∈ (ℤ‘3) ∧ 𝑃:(0...3)⟶𝑉) → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
7472, 73mpan 702 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑃:(0...3)⟶𝑉 → ∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
75 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑) → (𝑃‘2) = 𝑐)
7675anim2i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (𝑃‘2) = 𝑐))
77 df-3an 1033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) ↔ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ (𝑃‘2) = 𝑐))
7876, 77sylibr 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
7978rexlimivw 3011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
8079reximi 2994 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
8180reximi 2994 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑏𝑉𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
8281reximi 2994 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∃𝑎𝑉𝑏𝑉𝑐𝑉𝑑𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
8374, 82syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃:(0...3)⟶𝑉 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
84 preq12 4214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
85843adant3 1074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
8685eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ↔ {𝑎, 𝑏} ∈ ran 𝐸))
87 preq12 4214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘1), (𝑃‘2)} = {𝑏, 𝑐})
88873adant1 1072 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘1), (𝑃‘2)} = {𝑏, 𝑐})
8988eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ({(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ↔ {𝑏, 𝑐} ∈ ran 𝐸))
90 preq12 4214 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑃‘2) = 𝑐 ∧ (𝑃‘0) = 𝑎) → {(𝑃‘2), (𝑃‘0)} = {𝑐, 𝑎})
9190ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑃‘0) = 𝑎 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘2), (𝑃‘0)} = {𝑐, 𝑎})
92913adant2 1073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘2), (𝑃‘0)} = {𝑐, 𝑎})
9392eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ({(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸 ↔ {𝑐, 𝑎} ∈ ran 𝐸))
9486, 89, 933anbi123d 1391 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) ↔ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
9594biimpcd 238 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
9695reximdv 2999 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (∃𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ∃𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
9796reximdv 2999 . . . . . . . . . . . . . . . . . . . . . . . . 25 (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (∃𝑏𝑉𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
9897reximdv 2999 . . . . . . . . . . . . . . . . . . . . . . . 24 (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
9969, 83, 98syl2im 39 . . . . . . . . . . . . . . . . . . . . . . 23 ((((Fun 𝐸 ∧ (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹)) ∧ (#‘𝐹) = 3) ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)})) → (𝑃:(0...3)⟶𝑉 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
10099exp41 636 . . . . . . . . . . . . . . . . . . . . . 22 (Fun 𝐸 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → ((#‘𝐹) = 3 → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}) → (𝑃:(0...3)⟶𝑉 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
101100com14 94 . . . . . . . . . . . . . . . . . . . . 21 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → ((#‘𝐹) = 3 → (Fun 𝐸 → (𝑃:(0...3)⟶𝑉 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
102101com35 96 . . . . . . . . . . . . . . . . . . . 20 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → (𝑃:(0...3)⟶𝑉 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
10345, 102syl6bi 242 . . . . . . . . . . . . . . . . . . 19 ((𝑃‘0) = (𝑃‘3) → (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → (𝑃:(0...3)⟶𝑉 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
104103com12 32 . . . . . . . . . . . . . . . . . 18 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}) → ((𝑃‘0) = (𝑃‘3) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → (𝑃:(0...3)⟶𝑉 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
105104com24 93 . . . . . . . . . . . . . . . . 17 (((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}) → (𝑃:(0...3)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
10641, 105sylbi 206 . . . . . . . . . . . . . . . 16 (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → (𝑃:(0...3)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
107106com13 86 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) → (𝑃:(0...3)⟶𝑉 → (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
1081073imp 1249 . . . . . . . . . . . . . 14 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
109108com14 94 . . . . . . . . . . . . 13 ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
110 fveq2 6103 . . . . . . . . . . . . . 14 ((#‘𝐹) = 3 → (𝑃‘(#‘𝐹)) = (𝑃‘3))
111110eqeq2d 2620 . . . . . . . . . . . . 13 ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
112 oveq2 6557 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = 3 → (0...(#‘𝐹)) = (0...3))
113112feq2d 5944 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 3 → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...3)⟶𝑉))
114 oveq2 6557 . . . . . . . . . . . . . . . . 17 ((#‘𝐹) = 3 → (0..^(#‘𝐹)) = (0..^3))
115114raleqdv 3121 . . . . . . . . . . . . . . . 16 ((#‘𝐹) = 3 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
116113, 1153anbi23d 1394 . . . . . . . . . . . . . . 15 ((#‘𝐹) = 3 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
117116imbi1d 330 . . . . . . . . . . . . . 14 ((#‘𝐹) = 3 → ((((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) ↔ (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
118117imbi2d 329 . . . . . . . . . . . . 13 ((#‘𝐹) = 3 → ((Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))) ↔ (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
119109, 111, 1183imtr4d 282 . . . . . . . . . . . 12 ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
120119com14 94 . . . . . . . . . . 11 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
1211202a1d 26 . . . . . . . . . 10 (((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) → (Fun (𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
1226, 121syl6bi 242 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 → (Fun (𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))))
1231223impd 1273 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
1245, 123sylbid 229 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
125124impd 446 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
1264, 125sylbid 229 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
1271263adant1 1072 . . . 4 (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
1283, 127mpcom 37 . . 3 (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
129128com12 32 . 2 (Fun 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 3 → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
1301293imp 1249 1 ((Fun 𝐸𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539  ∅c0 3874  {cpr 4127  {ctp 4129   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  2c2 10947  3c3 10948  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   Walks cwalk 26026   Trails ctrail 26027   Paths cpath 26028   Cycles ccycl 26035 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 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-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-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  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-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-cycl 26041 This theorem is referenced by:  3v3e3cycl  26193
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