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Definition df-uslgra 25059
Description: Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph  <. V ,  E >. where  E is an injective (one-to-one) function into subsets of  V of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
df-uslgra  |- USLGrph  =  { <. 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-uslgra
StepHypRef Expression
1 cuslg 25056 . 2  class USLGrph
2 ve . . . . . 6  setvar  e
32cv 1443 . . . . 5  class  e
43cdm 4834 . . . 4  class  dom  e
5 vx . . . . . . . 8  setvar  x
65cv 1443 . . . . . . 7  class  x
7 chash 12515 . . . . . . 7  class  #
86, 7cfv 5582 . . . . . 6  class  ( # `  x )
9 c2 10659 . . . . . 6  class  2
10 cle 9676 . . . . . 6  class  <_
118, 9, 10wbr 4402 . . . . 5  wff  ( # `  x )  <_  2
12 vv . . . . . . . 8  setvar  v
1312cv 1443 . . . . . . 7  class  v
1413cpw 3951 . . . . . 6  class  ~P v
15 c0 3731 . . . . . . 7  class  (/)
1615csn 3968 . . . . . 6  class  { (/) }
1714, 16cdif 3401 . . . . 5  class  ( ~P v  \  { (/) } )
1811, 5, 17crab 2741 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }
194, 18, 3wf1 5579 . . 3  wff  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }
2019, 12, 2copab 4460 . 2  class  { <. v ,  e >.  |  e : dom  e -1-1-> {
x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
211, 20wceq 1444 1  wff USLGrph  =  { <. v ,  e >.  |  e : dom  e -1-1-> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
Colors of variables: wff setvar class
This definition is referenced by:  reluslgra  25061  isuslgra  25070
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