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Definition df-usgra 23288
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 23286 . 2  class USGrph
2 ve . . . . . 6  setvar  e
32cv 1368 . . . . 5  class  e
43cdm 4861 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1368 . . . . . . 7  class  x
7 chash 12124 . . . . . . 7  class  #
86, 7cfv 5439 . . . . . 6  class  ( # `  x )
9 c2 10392 . . . . . 6  class  2
108, 9wceq 1369 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  setvar  v
1211cv 1368 . . . . . . 7  class  v
1312cpw 3881 . . . . . 6  class  ~P v
14 c0 3658 . . . . . . 7  class  (/)
1514csn 3898 . . . . . 6  class  { (/) }
1613, 15cdif 3346 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2740 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5436 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4370 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1369 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  23290  isusgra  23294
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