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Definition df-grtri 47719
Description: Definition of a triangles in a graph. A triangle in a graph is a set of three (different) vertices completely connected with each other. Such vertices induce a closed walk of length 3, see grtriclwlk3 47726. (TODO: and a cycle of length 3 ,see grtricycl ). (Contributed by AV, 20-Jul-2025.)
Assertion
Ref Expression
df-grtri GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
Distinct variable group:   𝑒,𝑓,𝑔,𝑡,𝑣

Detailed syntax breakdown of Definition df-grtri
StepHypRef Expression
1 cgrtri 47718 . 2 class GrTriangles
2 vg . . 3 setvar 𝑔
3 cvv 3482 . . 3 class V
4 vv . . . 4 setvar 𝑣
52cv 1536 . . . . 5 class 𝑔
6 cvtx 29022 . . . . 5 class Vtx
75, 6cfv 6572 . . . 4 class (Vtx‘𝑔)
8 ve . . . . 5 setvar 𝑒
9 cedg 29073 . . . . . 6 class Edg
105, 9cfv 6572 . . . . 5 class (Edg‘𝑔)
11 cc0 11180 . . . . . . . . . 10 class 0
12 c3 12345 . . . . . . . . . 10 class 3
13 cfzo 13707 . . . . . . . . . 10 class ..^
1411, 12, 13co 7445 . . . . . . . . 9 class (0..^3)
15 vt . . . . . . . . . 10 setvar 𝑡
1615cv 1536 . . . . . . . . 9 class 𝑡
17 vf . . . . . . . . . 10 setvar 𝑓
1817cv 1536 . . . . . . . . 9 class 𝑓
1914, 16, 18wf1o 6571 . . . . . . . 8 wff 𝑓:(0..^3)–1-1-onto𝑡
2011, 18cfv 6572 . . . . . . . . . . 11 class (𝑓‘0)
21 c1 11181 . . . . . . . . . . . 12 class 1
2221, 18cfv 6572 . . . . . . . . . . 11 class (𝑓‘1)
2320, 22cpr 4650 . . . . . . . . . 10 class {(𝑓‘0), (𝑓‘1)}
248cv 1536 . . . . . . . . . 10 class 𝑒
2523, 24wcel 2103 . . . . . . . . 9 wff {(𝑓‘0), (𝑓‘1)} ∈ 𝑒
26 c2 12344 . . . . . . . . . . . 12 class 2
2726, 18cfv 6572 . . . . . . . . . . 11 class (𝑓‘2)
2820, 27cpr 4650 . . . . . . . . . 10 class {(𝑓‘0), (𝑓‘2)}
2928, 24wcel 2103 . . . . . . . . 9 wff {(𝑓‘0), (𝑓‘2)} ∈ 𝑒
3022, 27cpr 4650 . . . . . . . . . 10 class {(𝑓‘1), (𝑓‘2)}
3130, 24wcel 2103 . . . . . . . . 9 wff {(𝑓‘1), (𝑓‘2)} ∈ 𝑒
3225, 29, 31w3a 1087 . . . . . . . 8 wff ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)
3319, 32wa 395 . . . . . . 7 wff (𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))
3433, 17wex 1777 . . . . . 6 wff 𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))
354cv 1536 . . . . . . 7 class 𝑣
3635cpw 4622 . . . . . 6 class 𝒫 𝑣
3734, 15, 36crab 3438 . . . . 5 class {𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}
388, 10, 37csb 3915 . . . 4 class (Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}
394, 7, 38csb 3915 . . 3 class (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}
402, 3, 39cmpt 5252 . 2 class (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
411, 40wceq 1537 1 wff GrTriangles = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(Edg‘𝑔) / 𝑒{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))})
Colors of variables: wff setvar class
This definition is referenced by:  grtri  47721
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