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Definition df-usgra 24538
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 24535 . 2  class USGrph
2 ve . . . . . 6  setvar  e
32cv 1397 . . . . 5  class  e
43cdm 4988 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1397 . . . . . . 7  class  x
7 chash 12390 . . . . . . 7  class  #
86, 7cfv 5570 . . . . . 6  class  ( # `  x )
9 c2 10581 . . . . . 6  class  2
108, 9wceq 1398 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  setvar  v
1211cv 1397 . . . . . . 7  class  v
1312cpw 3999 . . . . . 6  class  ~P v
14 c0 3783 . . . . . . 7  class  (/)
1514csn 4016 . . . . . 6  class  { (/) }
1613, 15cdif 3458 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2808 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5567 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4496 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1398 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  24540  isusgra  24549  usgraop  24555
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