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Definition df-uvtxa 39567
Description: Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph) resp. all other vertices are its neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)
Assertion
Ref Expression
df-uvtxa  |- UnivVtx  =  ( g  e.  _V  |->  { v  e.  (Vtx `  g )  |  A. n  e.  ( (Vtx `  g )  \  {
v } ) n  e.  ( g NeighbVtx  v
) } )
Distinct variable group:    v, g, n

Detailed syntax breakdown of Definition df-uvtxa
StepHypRef Expression
1 cuvtxa 39562 . 2  class UnivVtx
2 vg . . 3  setvar  g
3 cvv 3031 . . 3  class  _V
4 vn . . . . . . 7  setvar  n
54cv 1451 . . . . . 6  class  n
62cv 1451 . . . . . . 7  class  g
7 vv . . . . . . . 8  setvar  v
87cv 1451 . . . . . . 7  class  v
9 cnbgr 39561 . . . . . . 7  class NeighbVtx
106, 8, 9co 6308 . . . . . 6  class  ( g NeighbVtx  v )
115, 10wcel 1904 . . . . 5  wff  n  e.  ( g NeighbVtx  v )
12 cvtx 39251 . . . . . . 7  class Vtx
136, 12cfv 5589 . . . . . 6  class  (Vtx `  g )
148csn 3959 . . . . . 6  class  { v }
1513, 14cdif 3387 . . . . 5  class  ( (Vtx
`  g )  \  { v } )
1611, 4, 15wral 2756 . . . 4  wff  A. n  e.  ( (Vtx `  g
)  \  { v } ) n  e.  ( g NeighbVtx  v )
1716, 7, 13crab 2760 . . 3  class  { v  e.  (Vtx `  g
)  |  A. n  e.  ( (Vtx `  g
)  \  { v } ) n  e.  ( g NeighbVtx  v ) }
182, 3, 17cmpt 4454 . 2  class  ( g  e.  _V  |->  { v  e.  (Vtx `  g
)  |  A. n  e.  ( (Vtx `  g
)  \  { v } ) n  e.  ( g NeighbVtx  v ) } )
191, 18wceq 1452 1  wff UnivVtx  =  ( g  e.  _V  |->  { v  e.  (Vtx `  g )  |  A. n  e.  ( (Vtx `  g )  \  {
v } ) n  e.  ( g NeighbVtx  v
) } )
Colors of variables: wff setvar class
This definition is referenced by:  uvtxaval  39623
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