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Definition df-usgra 23088
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 23086 . 2  class USGrph
2 ve . . . . . 6  setvar  e
32cv 1361 . . . . 5  class  e
43cdm 4827 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1361 . . . . . . 7  class  x
7 chash 12086 . . . . . . 7  class  #
86, 7cfv 5406 . . . . . 6  class  ( # `  x )
9 c2 10358 . . . . . 6  class  2
108, 9wceq 1362 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  setvar  v
1211cv 1361 . . . . . . 7  class  v
1312cpw 3848 . . . . . 6  class  ~P v
14 c0 3625 . . . . . . 7  class  (/)
1514csn 3865 . . . . . 6  class  { (/) }
1613, 15cdif 3313 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2709 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5403 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4337 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1362 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  23090  isusgra  23094
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