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Definition df-conngr 41354
 Description: Define the class of all connected graphs. A graph is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv-av 41297 and dfconngr1 41355. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)
Assertion
Ref Expression
df-conngr ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
Distinct variable group:   𝑣,𝑔,𝑘,𝑛,𝑓,𝑝

Detailed syntax breakdown of Definition df-conngr
StepHypRef Expression
1 cconngr 41353 . 2 class ConnGraph
2 vf . . . . . . . . . 10 setvar 𝑓
32cv 1474 . . . . . . . . 9 class 𝑓
4 vp . . . . . . . . . 10 setvar 𝑝
54cv 1474 . . . . . . . . 9 class 𝑝
6 vk . . . . . . . . . . 11 setvar 𝑘
76cv 1474 . . . . . . . . . 10 class 𝑘
8 vn . . . . . . . . . . 11 setvar 𝑛
98cv 1474 . . . . . . . . . 10 class 𝑛
10 vg . . . . . . . . . . . 12 setvar 𝑔
1110cv 1474 . . . . . . . . . . 11 class 𝑔
12 cpthson 40921 . . . . . . . . . . 11 class PathsOn
1311, 12cfv 5804 . . . . . . . . . 10 class (PathsOn‘𝑔)
147, 9, 13co 6549 . . . . . . . . 9 class (𝑘(PathsOn‘𝑔)𝑛)
153, 5, 14wbr 4583 . . . . . . . 8 wff 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝
1615, 4wex 1695 . . . . . . 7 wff 𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝
1716, 2wex 1695 . . . . . 6 wff 𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝
18 vv . . . . . . 7 setvar 𝑣
1918cv 1474 . . . . . 6 class 𝑣
2017, 8, 19wral 2896 . . . . 5 wff 𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝
2120, 6, 19wral 2896 . . . 4 wff 𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝
22 cvtx 25673 . . . . 5 class Vtx
2311, 22cfv 5804 . . . 4 class (Vtx‘𝑔)
2421, 18, 23wsbc 3402 . . 3 wff [(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝
2524, 10cab 2596 . 2 class {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
261, 25wceq 1475 1 wff ConnGraph = {𝑔[(Vtx‘𝑔) / 𝑣]𝑘𝑣𝑛𝑣𝑓𝑝 𝑓(𝑘(PathsOn‘𝑔)𝑛)𝑝}
 Colors of variables: wff setvar class This definition is referenced by:  dfconngr1  41355  isconngr  41356
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