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Definition df-2spthsot 28058
Description: Define the collection of all simple paths of length 2 as ordered triple. (in a graph) (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
df-2spthsot  |- 2SPathOnOt  =  ( v  e.  _V , 
e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e )
b ) } )
Distinct variable group:    a, b, e, v, t

Detailed syntax breakdown of Definition df-2spthsot
StepHypRef Expression
1 c2spthot 28053 . 2  class 2SPathOnOt
2 vv . . 3  set  v
3 ve . . 3  set  e
4 cvv 2916 . . 3  class  _V
5 vt . . . . . . . 8  set  t
65cv 1648 . . . . . . 7  class  t
7 va . . . . . . . . 9  set  a
87cv 1648 . . . . . . . 8  class  a
9 vb . . . . . . . . 9  set  b
109cv 1648 . . . . . . . 8  class  b
112cv 1648 . . . . . . . . 9  class  v
123cv 1648 . . . . . . . . 9  class  e
13 c2pthonot 28054 . . . . . . . . 9  class 2SPathOnOt
1411, 12, 13co 6040 . . . . . . . 8  class  ( v 2SPathOnOt  e )
158, 10, 14co 6040 . . . . . . 7  class  ( a ( v 2SPathOnOt  e )
b )
166, 15wcel 1721 . . . . . 6  wff  t  e.  ( a ( v 2SPathOnOt  e ) b )
1716, 9, 11wrex 2667 . . . . 5  wff  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e ) b )
1817, 7, 11wrex 2667 . . . 4  wff  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e ) b )
1911, 11cxp 4835 . . . . 5  class  ( v  X.  v )
2019, 11cxp 4835 . . . 4  class  ( ( v  X.  v )  X.  v )
2118, 5, 20crab 2670 . . 3  class  { t  e.  ( ( v  X.  v )  X.  v )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e ) b ) }
222, 3, 4, 4, 21cmpt2 6042 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e ) b ) } )
231, 22wceq 1649 1  wff 2SPathOnOt  =  ( v  e.  _V , 
e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e )
b ) } )
Colors of variables: wff set class
This definition is referenced by:  2spthsot  28065
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