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Definition df-usgra 25058
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 25055 . 2  class USGrph
2 ve . . . . . 6  setvar  e
32cv 1436 . . . . 5  class  e
43cdm 4853 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1436 . . . . . . 7  class  x
7 chash 12521 . . . . . . 7  class  #
86, 7cfv 5601 . . . . . 6  class  ( # `  x )
9 c2 10666 . . . . . 6  class  2
108, 9wceq 1437 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  setvar  v
1211cv 1436 . . . . . . 7  class  v
1312cpw 3981 . . . . . 6  class  ~P v
14 c0 3761 . . . . . . 7  class  (/)
1514csn 3998 . . . . . 6  class  { (/) }
1613, 15cdif 3433 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2775 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5598 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4481 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1437 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  25060  isusgra  25069  usgraop  25075
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