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Theorem el2spthonot 25533
Description: A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
el2spthonot  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
Distinct variable groups:    A, b,
f, p    C, b,
f, p    E, b,
f, p    T, b,
f, p    V, b,
f, p    X, b,
f, p    Y, b,
f, p

Proof of Theorem el2spthonot
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 2spthonot 25529 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) C )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } )
21eleq2d 2485 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  T  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } ) )
3 fveq2 5818 . . . . . . . . 9  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
43fveq2d 5822 . . . . . . . 8  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
54eqeq1d 2424 . . . . . . 7  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  T
) )  =  A ) )
63fveq2d 5822 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
76eqeq1d 2424 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
8 fveq2 5818 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
98eqeq1d 2424 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd `  T )  =  C ) )
105, 7, 93anbi123d 1335 . . . . . 6  |-  ( t  =  T  ->  (
( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C )  <-> 
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )
11103anbi3d 1341 . . . . 5  |-  ( t  =  T  ->  (
( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) )  <->  ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
12112exbidv 1764 . . . 4  |-  ( t  =  T  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  C ) )  <->  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
1312elrab 3164 . . 3  |-  ( T  e.  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) }  <->  ( T  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
142, 13syl6bb 264 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  ( T  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) ) )
15 vex 3019 . . . . . . . . . 10  |-  f  e. 
_V
16 vex 3019 . . . . . . . . . 10  |-  p  e. 
_V
1715, 16pm3.2i 456 . . . . . . . . 9  |-  ( f  e.  _V  /\  p  e.  _V )
18 isspthon 25248 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( f  e.  _V  /\  p  e. 
_V )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( f
( A ( V SPathOn  E ) C ) p  <->  ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p ) ) )
1917, 18mp3an2 1348 . . . . . . . 8  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( f ( A ( V SPathOn  E ) C ) p  <->  ( f
( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p ) ) )
20193anbi1d 1339 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) )  <->  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) )
2120anbi2d 708 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( f
( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  <-> 
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) ) )
22 simpl 458 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
2322ad2antll 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  A  e.  V )
24 spthispth 25238 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V SPaths  E ) p  ->  f ( V Paths  E ) p )
25 pthistrl 25237 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
26 trliswlk 25204 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
27 el2wlkonotlem 25525 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
2827ex 435 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
2924, 25, 26, 284syl 19 . . . . . . . . . . . . . . . . . 18  |-  ( f ( V SPaths  E ) p  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
3029adantl 467 . . . . . . . . . . . . . . . . 17  |-  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  ->  (
( # `  f )  =  2  ->  (
p `  1 )  e.  V ) )
3130imp 430 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E
) p )  /\  ( # `  f )  =  2 )  -> 
( p `  1
)  e.  V )
3231adantr 466 . . . . . . . . . . . . . . 15  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  (
p `  1 )  e.  V )
33 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3433ad2antll 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  C  e.  V )
35 el2xptp0 6788 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  ( p `  1
)  e.  V  /\  C  e.  V )  ->  ( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) )  <->  T  =  <. A ,  ( p `
 1 ) ,  C >. ) )
3623, 32, 34, 35syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  (
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) )  <->  T  =  <. A ,  ( p `
 1 ) ,  C >. ) )
37 oteq2 4133 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( p `  1 )  =  b  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
3837eqcoms 2430 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( p ` 
1 )  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
3938eqeq2d 2432 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  <->  T  =  <. A ,  b ,  C >. )
)
4039biimpd 210 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  ->  T  =  <. A , 
b ,  C >. ) )
4140adantl 467 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) ) )  /\  b  =  ( p `  1 ) )  ->  ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  ->  T  = 
<. A ,  b ,  C >. ) )
42 simpllr 767 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  f
( V SPaths  E )
p )
4342adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) ) )  /\  b  =  ( p `  1 ) )  ->  f ( V SPaths  E ) p )
44 simpllr 767 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) ) )  /\  b  =  ( p `  1 ) )  ->  ( # `  f
)  =  2 )
45 iswlkon 25197 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( f  e.  _V  /\  p  e. 
_V )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( f
( A ( V WalkOn  E ) C ) p  <->  ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C ) ) )
4617, 45mp3an2 1348 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( f ( A ( V WalkOn  E ) C ) p  <->  ( f
( V Walks  E )
p  /\  ( p `  0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C ) ) )
47 fveq2 5818 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
4847eqeq1d 2424 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
4948anbi2d 708 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  f )  =  2  ->  (
( ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  <-> 
( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C ) ) )
50 eqcom 2429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( p `  0 )  =  A  <->  A  =  ( p `  0
) )
5150biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
5251adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  =  ( p `  0 ) )
5352adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  A  =  ( p ` 
0 ) )
54 simpr 462 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  b  =  ( p ` 
1 ) )
55 eqcom 2429 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( p `  2 )  =  C  <->  C  =  ( p `  2
) )
5655biimpi 197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
5756adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  C  =  ( p `  2 ) )
5857adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  C  =  ( p ` 
2 ) )
5953, 54, 583jca 1185 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )
6059ex 435 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
6160a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  f )  =  2  ->  (
( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  ->  ( b  =  ( p ` 
1 )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
6249, 61sylbid 218 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  f )  =  2  ->  (
( ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
6362com12 32 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  -> 
( ( # `  f
)  =  2  -> 
( b  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
64633adant1 1023 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( ( # `
 f )  =  2  ->  ( b  =  ( p ` 
1 )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
6564a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  ->  ( ( # `  f )  =  2  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
6646, 65sylbid 218 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( f ( A ( V WalkOn  E ) C ) p  -> 
( ( # `  f
)  =  2  -> 
( b  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
6766com3l 84 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( A ( V WalkOn  E ) C ) p  ->  ( ( # `
 f )  =  2  ->  ( (
( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( b  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
6867adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  ->  (
( # `  f )  =  2  ->  (
( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( b  =  ( p ` 
1 )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
6968imp41 596 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) ) )  /\  b  =  ( p `  1 ) )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )
7043, 44, 693jca 1185 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) ) )  /\  b  =  ( p `  1 ) )  ->  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
7141, 70jctird 546 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) ) )  /\  b  =  ( p `  1 ) )  ->  ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  ->  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
7232, 71rspcimedv 3120 . . . . . . . . . . . . . 14  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
7336, 72sylbid 218 . . . . . . . . . . . . 13  |-  ( ( ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2 )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  (
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
7473exp4b 610 . . . . . . . . . . . 12  |-  ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E
) p )  /\  ( # `  f )  =  2 )  -> 
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( T  e.  ( ( V  X.  V )  X.  V
)  ->  ( (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) )
7574com24 90 . . . . . . . . . . 11  |-  ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E
) p )  /\  ( # `  f )  =  2 )  -> 
( ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C )  ->  ( T  e.  ( ( V  X.  V )  X.  V )  ->  (
( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) ) ) )
7675ex 435 . . . . . . . . . 10  |-  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  ->  (
( # `  f )  =  2  ->  (
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C )  ->  ( T  e.  ( ( V  X.  V )  X.  V
)  ->  ( (
( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) ) )
77763imp 1199 . . . . . . . . 9  |-  ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E
) p )  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) )  ->  ( T  e.  ( ( V  X.  V )  X.  V
)  ->  ( (
( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) )
7877impcom 431 . . . . . . . 8  |-  ( ( T  e.  ( ( V  X.  V )  X.  V )  /\  ( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )  ->  (
( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
7978com12 32 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( (
f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  /\  ( # `
 f )  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
8022adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
81 simpr 462 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
8233adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
8380, 81, 823jca 1185 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
84 otel3xp 4825 . . . . . . . . . . . . . . . . 17  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V
) )  ->  T  e.  ( ( V  X.  V )  X.  V
) )
8583, 84sylan2 476 . . . . . . . . . . . . . . . 16  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) )
8685ex 435 . . . . . . . . . . . . . . 15  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) )
8786adantr 466 . . . . . . . . . . . . . 14  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  (
( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) )
8887com12 32 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) ) )
8988ex 435 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  C  e.  V )  ->  ( b  e.  V  ->  ( ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) ) )
9089adantl 467 . . . . . . . . . . 11  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( b  e.  V  ->  ( ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) ) )
9190imp31 433 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) )
9224, 25, 263syl 18 . . . . . . . . . . . . . . 15  |-  ( f ( V SPaths  E ) p  ->  f ( V Walks  E ) p )
93923ad2ant1 1026 . . . . . . . . . . . . . 14  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  f ( V Walks 
E ) p )
9493ad2antll 733 . . . . . . . . . . . . 13  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
f ( V Walks  E
) p )
95 eqcom 2429 . . . . . . . . . . . . . . . . 17  |-  ( A  =  ( p ` 
0 )  <->  ( p `  0 )  =  A )
9695biimpi 197 . . . . . . . . . . . . . . . 16  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
97963ad2ant1 1026 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
98973ad2ant3 1028 . . . . . . . . . . . . . 14  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
9998ad2antll 733 . . . . . . . . . . . . 13  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( p `  0
)  =  A )
100 eqcom 2429 . . . . . . . . . . . . . . . . . . 19  |-  ( C  =  ( p ` 
2 )  <->  ( p `  2 )  =  C )
101100biimpi 197 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
1021013ad2ant3 1028 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
103102, 48syl5ibr 224 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p `  ( # `  f
) )  =  C ) )
104103a1i 11 . . . . . . . . . . . . . . 15  |-  ( f ( V SPaths  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p `  ( # `  f ) )  =  C ) ) )
1051043imp 1199 . . . . . . . . . . . . . 14  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
106105ad2antll 733 . . . . . . . . . . . . 13  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( p `  ( # `
 f ) )  =  C )
10746adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
f ( A ( V WalkOn  E ) C ) p  <->  ( f
( V Walks  E )
p  /\  ( p `  0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C ) ) )
108107adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( f ( A ( V WalkOn  E ) C ) p  <->  ( f
( V Walks  E )
p  /\  ( p `  0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C ) ) )
10994, 99, 106, 108mpbir3and 1188 . . . . . . . . . . . 12  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
f ( A ( V WalkOn  E ) C ) p )
110 id 22 . . . . . . . . . . . . . 14  |-  ( f ( V SPaths  E ) p  ->  f ( V SPaths  E ) p )
1111103ad2ant1 1026 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  f ( V SPaths  E ) p )
112111ad2antll 733 . . . . . . . . . . . 12  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
f ( V SPaths  E
) p )
113109, 112jca 534 . . . . . . . . . . 11  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E
) p ) )
114 id 22 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  2  ->  ( # `
 f )  =  2 )
1151143ad2ant2 1027 . . . . . . . . . . . 12  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( # `  f
)  =  2 )
116115ad2antll 733 . . . . . . . . . . 11  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( # `  f )  =  2 )
11722adantl 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
118117adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  A  e.  V )
119 simpr 462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  b  e.  V )
12033adantl 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
121120adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  C  e.  V )
122118, 119, 1213jca 1185 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
123 oteqimp 6763 . . . . . . . . . . . . . . . . . . 19  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( A  e.  V  /\  b  e.  V  /\  C  e.  V
)  ->  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  b  /\  ( 2nd `  T )  =  C ) ) )
124123imp 430 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V
) )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  b  /\  ( 2nd `  T
)  =  C ) )
125122, 124sylan2 476 . . . . . . . . . . . . . . . . 17  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V ) )  -> 
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  b  /\  ( 2nd `  T
)  =  C ) )
126125ex 435 . . . . . . . . . . . . . . . 16  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  b  /\  ( 2nd `  T
)  =  C ) ) )
127 eqeq2 2433 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  1 )  =  b  ->  (
( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  b ) )
128127eqcoms 2430 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  b ) )
1291283anbi2d 1340 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( p ` 
1 )  ->  (
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C )  <-> 
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  b  /\  ( 2nd `  T
)  =  C ) ) )
130129imbi2d 317 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( p ` 
1 )  ->  (
( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) )  <->  ( ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  /\  b  e.  V )  ->  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  b  /\  ( 2nd `  T
)  =  C ) ) ) )
131126, 130syl5ibr 224 . . . . . . . . . . . . . . 15  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
b ,  C >.  -> 
( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V
) )  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
1321313ad2ant2 1027 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( T  = 
<. A ,  b ,  C >.  ->  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
1331323ad2ant3 1028 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( T  = 
<. A ,  b ,  C >.  ->  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
134133impcom 431 . . . . . . . . . . . 12  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  (
( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )
135134impcom 431 . . . . . . . . . . 11  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) )
136113, 116, 1353jca 1185 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( ( f ( A ( V WalkOn  E
) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )
13791, 136jca 534 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) )
138137ex 435 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  -> 
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E ) p )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) ) )
139138rexlimdva 2850 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( E. b  e.  V  ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( (
f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  /\  ( # `
 f )  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) ) )
14079, 139impbid 193 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( (
f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  /\  ( # `
 f )  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
14121, 140bitrd 256 . . . . 5  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( f
( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
1421412exbidv 1764 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( E. f E. p ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( f
( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  <->  E. f E. p E. b  e.  V  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
143 19.42vv 1829 . . . 4  |-  ( E. f E. p ( T  e.  ( ( V  X.  V )  X.  V )  /\  ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )  <->  ( T  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
144 rexcom4 3037 . . . . 5  |-  ( E. b  e.  V  E. f E. p ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  <->  E. f E. b  e.  V  E. p
( T  =  <. A ,  b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
145 rexcom4 3037 . . . . . 6  |-  ( E. b  e.  V  E. p ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  <->  E. p E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
146145exbii 1712 . . . . 5  |-  ( E. f E. b  e.  V  E. p ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  E. f E. p E. b  e.  V  ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
147144, 146bitr2i 253 . . . 4  |-  ( E. f E. p E. b  e.  V  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  E. b  e.  V  E. f E. p ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
148142, 143, 1473bitr3g 290 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )  <->  E. b  e.  V  E. f E. p ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
149 19.42vv 1829 . . . 4  |-  ( E. f E. p ( T  =  <. A , 
b ,  C >.  /\  ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
150149rexbii 2860 . . 3  |-  ( E. b  e.  V  E. f E. p ( T  =  <. A ,  b ,  C >.  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
151148, 150syl6bb 264 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
15214, 151bitrd 256 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   E.wrex 2709   {crab 2712   _Vcvv 3016   <.cotp 3942   class class class wbr 4359    X. cxp 4787   ` cfv 5537  (class class class)co 6242   1stc1st 6742   2ndc2nd 6743   0cc0 9483   1c1 9484   2c2 10603   #chash 12458   Walks cwalk 25161   Trails ctrail 25162   Paths cpath 25163   SPaths cspath 25164   WalkOn cwlkon 25165   SPathOn cspthon 25168   2SPathOnOt c2pthonot 25520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-ot 3943  df-uni 4156  df-int 4192  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-om 6644  df-1st 6744  df-2nd 6745  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8318  df-cda 8542  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-nn 10554  df-2 10612  df-n0 10814  df-z 10882  df-uz 11104  df-fz 11729  df-fzo 11860  df-hash 12459  df-word 12605  df-wlk 25171  df-trail 25172  df-pth 25173  df-spth 25174  df-wlkon 25177  df-spthon 25180  df-2spthonot 25523
This theorem is referenced by:  el2spthonot0  25534  el2pthsot  25544  2spontn0vne  25550  2spotiundisj  25725
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