3v3e3cycl1 - Metamath Proof Explorer

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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 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({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.

Step	Hyp	Ref	Expression
1		cycliswlk 26160	. . . . 5 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → 𝐹(𝑉 Walks 𝐸)𝑃)
2		wlkbprop 26051	. . . . 5 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3	1, 2	syl 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‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
7		fzo0to3tp 12421	. . . . . . . . . . . . . . . . . . 19 ⊢ (0..^3) = {0, 1, 2}
8	7	raleqi 3119	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (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 → (𝐹‘𝑘) = (𝐹‘0))
13	12	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘0)))
14		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
15		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
16		0p1e1 11009	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 + 1) = 1
17	15, 16	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
18	17	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → (𝑃‘(𝑘 + 1)) = (𝑃‘1))
19	14, 18	preq12d 4220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘0), (𝑃‘1)})
20	13, 19	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)}))
21		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1))
22	21	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 1 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘1)))
23		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 1 → (𝑃‘𝑘) = (𝑃‘1))
24		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 1 → (𝑘 + 1) = (1 + 1))
25		1p1e2 11011	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 + 1) = 2
26	24, 25	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 1 → (𝑘 + 1) = 2)
27	26	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 1 → (𝑃‘(𝑘 + 1)) = (𝑃‘2))
28	23, 27	preq12d 4220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 1 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘1), (𝑃‘2)})
29	22, 28	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 1 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
30		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → (𝐹‘𝑘) = (𝐹‘2))
31	30	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 2 → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘2)))
32		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → (𝑃‘𝑘) = (𝑃‘2))
33		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 2 → (𝑘 + 1) = (2 + 1))
34		2p1e3 11028	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (2 + 1) = 3
35	33, 34	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 2 → (𝑘 + 1) = 3)
36	35	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 2 → (𝑃‘(𝑘 + 1)) = (𝑃‘3))
37	32, 36	preq12d 4220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 2 → {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} = {(𝑃‘2), (𝑃‘3)})
38	31, 37	eqeq12d 2625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 2 → ((𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
39	20, 29, 38	raltpg 4183	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 2 ∈ ℤ) → (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)})))
40	9, 10, 11, 39	mp3an 1416	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑘 ∈ {0, 1, 2} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} ∧ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)}))
41	8, 40	bitri 263	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (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)})
43	42	eqcoms 2618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑃‘0) = (𝑃‘3) → {(𝑃‘2), (𝑃‘3)} = {(𝑃‘2), (𝑃‘0)})
44	43	eqeq2d 2620	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘0) = (𝑃‘3) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘3)} ↔ (𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)}))
45	44	3anbi3d 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))
48	46, 47	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 3 → 0 < (#‘𝐹))
49		0nn0 11184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ∈ ℕ₀
50	48, 49	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐹) = 3 → (0 ∈ ℕ₀ ∧ 0 < (#‘𝐹)))
51		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (0 ∈ ℕ₀ ∧ 0 < (#‘𝐹))) → ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} → {(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸))
52	50, 51	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ 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))
55	53, 54	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 3 → 1 < (#‘𝐹))
56		1nn0 11185	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 ∈ ℕ₀
57	55, 56	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐹) = 3 → (1 ∈ ℕ₀ ∧ 1 < (#‘𝐹)))
58		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (1 ∈ ℕ₀ ∧ 1 < (#‘𝐹))) → ((𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)} → {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸))
59	57, 58	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ 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))
62	60, 61	mpbiri 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((#‘𝐹) = 3 → 2 < (#‘𝐹))
63		2nn0 11186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 2 ∈ ℕ₀
64	62, 63	jctil 558	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝐹) = 3 → (2 ∈ ℕ₀ ∧ 2 < (#‘𝐹)))
65		nvnencycllem 26171	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (2 ∈ ℕ₀ ∧ 2 < (#‘𝐹))) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)} → {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸))
66	64, 65	sylan2 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (#‘𝐹) = 3) → ((𝐸‘(𝐹‘2)) = {(𝑃‘2), (𝑃‘0)} → {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸))
67	52, 59, 66	3anim123d 1398	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((Fun 𝐸 ∧ 𝐹 ∈ 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)))
68	67	adantlrr 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)))
69	68	imp 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))
72	70, 71	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 3 ∈ (ℤ_≥‘3)
73		4fvwrd4 12328	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((3 ∈ (ℤ_≥‘3) ∧ 𝑃:(0...3)⟶𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)))
74	72, 73	mpan 702	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃:(0...3)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((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)
76	75	anim2i 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))
78	76, 77	sylibr 223	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
79	78	rexlimivw 3011	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
80	79	reximi 2994	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑐 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
81	80	reximi 2994	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
82	81	reximi 2994	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ∃𝑑 ∈ 𝑉 (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏) ∧ ((𝑃‘2) = 𝑐 ∧ (𝑃‘3) = 𝑑)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐))
83	74, 82	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃:(0...3)⟶𝑉 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((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})
85	84	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘0), (𝑃‘1)} = {𝑎, 𝑏})
86	85	eleq1d 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})
88	87	3adant1 1072	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘1), (𝑃‘2)} = {𝑏, 𝑐})
89	88	eleq1d 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})
91	90	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘2), (𝑃‘0)} = {𝑐, 𝑎})
92	91	3adant2 1073	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → {(𝑃‘2), (𝑃‘0)} = {𝑐, 𝑎})
93	92	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ({(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸 ↔ {𝑐, 𝑎} ∈ ran 𝐸))
94	86, 89, 93	3anbi123d 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)))
95	94	biimpcd 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)))
96	95	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (∃𝑐 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
97	96	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
98	97	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (({(𝑃‘0), (𝑃‘1)} ∈ ran 𝐸 ∧ {(𝑃‘1), (𝑃‘2)} ∈ ran 𝐸 ∧ {(𝑃‘2), (𝑃‘0)} ∈ ran 𝐸) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑃‘0) = 𝑎 ∧ (𝑃‘1) = 𝑏 ∧ (𝑃‘2) = 𝑐) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
99	69, 83, 98	syl2im 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))
100	99	exp41 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
101	100	com14 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
102	101	com35 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
103	45, 102	syl6bi 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
104	103	com12 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
105	104	com24 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
106	41, 105	sylbi 206	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → (𝑃:(0...3)⟶𝑉 → ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
107	106	com13 86	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) → (𝑃:(0...3)⟶𝑉 → (∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
108	107	3imp 1249	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
109	108	com14 94	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘3) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
110		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 3 → (𝑃‘(#‘𝐹)) = (𝑃‘3))
111	110	eqeq2d 2620	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) ↔ (𝑃‘0) = (𝑃‘3)))
112		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 3 → (0...(#‘𝐹)) = (0...3))
113	112	feq2d 5944	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 3 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...3)⟶𝑉))
114		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝐹) = 3 → (0..^(#‘𝐹)) = (0..^3))
115	114	raleqdv 3121	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝐹) = 3 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
116	113, 115	3anbi23d 1394	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐹) = 3 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...3)⟶𝑉 ∧ ∀𝑘 ∈ (0..^3)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
117	116	imbi1d 330	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) = 3 → ((((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({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‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
118	117	imbi2d 329	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 3 → ((Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({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‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
119	109, 111, 118	3imtr4d 282	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) = 3 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
120	119	com14 94	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
121	120	2a1d 26	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word dom 𝐸 ∧ Fun ^◡𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) → (Fun ^◡(𝑃 ↾ (1..^(#‘𝐹))) → (((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅ → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))))
122	6, 121	syl6bi 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))))
123	122	3impd 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 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
124	5, 123	sylbid 229	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Paths 𝐸)𝑃 → ((𝑃‘0) = (𝑃‘(#‘𝐹)) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))))
125	124	impd 446	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝐹(𝑉 Paths 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
126	4, 125	sylbid 229	. . . . 5 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
127	126	3adant1 1072	. . . 4 ⊢ (((#‘𝐹) ∈ ℕ₀ ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)))))
128	3, 127	mpcom 37	. . 3 ⊢ (𝐹(𝑉 Cycles 𝐸)𝑃 → (Fun 𝐸 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
129	128	com12 32	. 2 ⊢ (Fun 𝐸 → (𝐹(𝑉 Cycles 𝐸)𝑃 → ((#‘𝐹) = 3 → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))))
130	129	3imp 1249	1 ⊢ ((Fun 𝐸 ∧ 𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({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

W3C validator