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Definition df-usgra 24311
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 24308 . 2  class USGrph
2 ve . . . . . 6  setvar  e
32cv 1382 . . . . 5  class  e
43cdm 4989 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1382 . . . . . . 7  class  x
7 chash 12387 . . . . . . 7  class  #
86, 7cfv 5578 . . . . . 6  class  ( # `  x )
9 c2 10592 . . . . . 6  class  2
108, 9wceq 1383 . . . . 5  wff  ( # `  x )  =  2
11 vv . . . . . . . 8  setvar  v
1211cv 1382 . . . . . . 7  class  v
1312cpw 3997 . . . . . 6  class  ~P v
14 c0 3770 . . . . . . 7  class  (/)
1514csn 4014 . . . . . 6  class  { (/) }
1613, 15cdif 3458 . . . . 5  class  ( ~P v  \  { (/) } )
1710, 5, 16crab 2797 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
184, 17, 3wf1 5575 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  =  2 }
1918, 11, 2copab 4494 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  =  2 } }
201, 19wceq 1383 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  24313  isusgra  24322  usgraop  24328
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