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Definition df-usgra 24037
Description: Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph  <. V ,  E >. where 
E is an injective (one-to-one) function into subsets of  V of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
df-usgra  |- USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } }
Distinct variable group:    v, e, x

Detailed syntax breakdown of Definition df-usgra
StepHypRef Expression
1 cusg 24034 . 2  class USGrph
2 ve . . . . . 6  setvar  e
32cv 1378 . . . . 5  class  e
43cdm 4999 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1378 . . . . . . 7  class  x
7 chash 12373 . . . . . . 7  class  #
86, 7cfv 5588 . . . . . 6  class  ( # `  x )
9 c2 10585 . . . . . 6  class  2
108, 9wceq 1379 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  setvar  v
1211cv 1378 . . . . . . 7  class  v
1312cpw 4010 . . . . . 6  class  ~P v
14 c0 3785 . . . . . . 7  class  (/)
1514csn 4027 . . . . . 6  class  { (/) }
1613, 15cdif 3473 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2818 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5585 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4504 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1379 1  wff USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } }
Colors of variables: wff setvar class
This definition is referenced by:  relusgra  24039  isusgra  24048  usgraop  24054
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