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Definition df-umgra 23878
Description: Define the class of all undirected multigraphs. A multigraph is a pair  <. V ,  E >. where  E is a 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. (Contributed by Mario Carneiro, 11-Mar-2015.)
Assertion
Ref Expression
df-umgra  |- UMGrph  =  { <. v ,  e >.  |  e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
Distinct variable group:    v, e, x

Detailed syntax breakdown of Definition df-umgra
StepHypRef Expression
1 cumg 23875 . 2  class UMGrph
2 ve . . . . . 6  set  e
32cv 1631 . . . . 5  class  e
43cdm 4705 . . . 4  class  dom  e
5 vx . . . . . . . 8  set  x
65cv 1631 . . . . . . 7  class  x
7 chash 11353 . . . . . . 7  class  #
86, 7cfv 5271 . . . . . 6  class  ( # `  x )
9 c2 9811 . . . . . 6  class  2
10 cle 8884 . . . . . 6  class  <_
118, 9, 10wbr 4039 . . . . 5  wff  ( # `  x )  <_  2
12 vv . . . . . . . 8  set  v
1312cv 1631 . . . . . . 7  class  v
1413cpw 3638 . . . . . 6  class  ~P v
15 c0 3468 . . . . . . 7  class  (/)
1615csn 3653 . . . . . 6  class  { (/) }
1714, 16cdif 3162 . . . . 5  class  ( ~P v  \  { (/) } )
1811, 5, 17crab 2560 . . . 4  class  { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }
194, 18, 3wf 5267 . . 3  wff  e : dom  e --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x
)  <_  2 }
2019, 12, 2copab 4092 . 2  class  { <. v ,  e >.  |  e : dom  e --> { x  e.  ( ~P v  \  { (/) } )  |  ( # `  x )  <_  2 } }
211, 20wceq 1632 1  wff UMGrph  =  { <. v ,  e >.  |  e : dom  e
--> { x  e.  ( ~P v  \  { (/)
} )  |  (
# `  x )  <_  2 } }
Colors of variables: wff set class
This definition is referenced by:  relumgra  23881  isumgra  23882
  Copyright terms: Public domain W3C validator