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Definition df-usgra 21367
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 itsself). (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 21365 . 2  class USGrph
2 ve . . . . . 6  set  e
32cv 1651 . . . . 5  class  e
43cdm 4878 . . . 4  class  dom  e
5 vx . . . . . . . 8  set  x
65cv 1651 . . . . . . 7  class  x
7 chash 11618 . . . . . . 7  class  #
86, 7cfv 5454 . . . . . 6  class  ( # `  x )
9 c2 10049 . . . . . 6  class  2
108, 9wceq 1652 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  set  v
1211cv 1651 . . . . . . 7  class  v
1312cpw 3799 . . . . . 6  class  ~P v
14 c0 3628 . . . . . . 7  class  (/)
1514csn 3814 . . . . . 6  class  { (/) }
1613, 15cdif 3317 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2709 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5451 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4265 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1652 1  wff USGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  =  2 } }
Colors of variables: wff set class
This definition is referenced by:  relusgra  21369  isusgra  21373
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