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Theorem el2spthonot 30394
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 30390 . . . 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 2510 . . 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 5696 . . . . . . . . 9  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
43fveq2d 5700 . . . . . . . 8  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
54eqeq1d 2451 . . . . . . 7  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  T
) )  =  A ) )
63fveq2d 5700 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
76eqeq1d 2451 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
8 fveq2 5696 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
98eqeq1d 2451 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd `  T )  =  C ) )
105, 7, 93anbi123d 1289 . . . . . 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 1295 . . . . 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 1682 . . . 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 3122 . . 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 261 . 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 2980 . . . . . . . . . 10  |-  f  e. 
_V
16 vex 2980 . . . . . . . . . 10  |-  p  e. 
_V
1715, 16pm3.2i 455 . . . . . . . . 9  |-  ( f  e.  _V  /\  p  e.  _V )
18 isspthon 23487 . . . . . . . . 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 1302 . . . . . . . 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 1293 . . . . . . 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 703 . . . . . 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 457 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
2322ad2antll 728 . . . . . . . . . . . . . . 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 23477 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V SPaths  E ) p  ->  f ( V Paths  E ) p )
25 pthistrl 23476 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
26 trliswlk 23443 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
27 el2wlkonotlem 30386 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
2827ex 434 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
2924, 25, 26, 284syl 21 . . . . . . . . . . . . . . . . . 18  |-  ( f ( V SPaths  E ) p  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
3029adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( ( f ( A ( V WalkOn  E ) C ) p  /\  f
( V SPaths  E )
p )  ->  (
( # `  f )  =  2  ->  (
p `  1 )  e.  V ) )
3130imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( A ( V WalkOn  E ) C ) p  /\  f ( V SPaths  E
) p )  /\  ( # `  f )  =  2 )  -> 
( p `  1
)  e.  V )
3231adantr 465 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3433ad2antll 728 . . . . . . . . . . . . . . 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 30132 . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . 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 4074 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( p `  1 )  =  b  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
3837eqcoms 2446 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( p ` 
1 )  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
3938eqeq2d 2454 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  <->  T  =  <. A ,  b ,  C >. )
)
4039biimpd 207 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  ->  T  =  <. A , 
b ,  C >. ) )
4140adantl 466 . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . 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 23435 . . . . . . . . . . . . . . . . . . . . . 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 1302 . . . . . . . . . . . . . . . . . . . . 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 5696 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
4847eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
4948anbi2d 703 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  f )  =  2  ->  (
( ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  <-> 
( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C ) ) )
50 eqcom 2445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( p `  0 )  =  A  <->  A  =  ( p `  0
) )
5150biimpi 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
5251adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  =  ( p `  0 ) )
5352adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  A  =  ( p ` 
0 ) )
54 simpr 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  b  =  ( p ` 
1 ) )
55 eqcom 2445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( p `  2 )  =  C  <->  C  =  ( p `  2
) )
5655biimpi 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
5756adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  C  =  ( p `  2 ) )
5857adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  C  =  ( p ` 
2 ) )
5953, 54, 583jca 1168 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )
6059ex 434 . . . . . . . . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  f )  =  2  ->  (
( ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
6362com12 31 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  -> 
( ( # `  f
)  =  2  -> 
( b  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
64633adant1 1006 . . . . . . . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . 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 544 . . . . . . . . . . . . . . 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 3080 . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . 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 607 . . . . . . . . . . . 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 87 . . . . . . . . . . 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 434 . . . . . . . . . 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 1181 . . . . . . . . 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 430 . . . . . . . 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 31 . . . . . . 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 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
81 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
8233adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
8380, 81, 823jca 1168 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
84 otel3xp 30133 . . . . . . . . . . . . . . . . 17  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V
) )  ->  T  e.  ( ( V  X.  V )  X.  V
) )
8583, 84sylan2 474 . . . . . . . . . . . . . . . 16  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) )
8685ex 434 . . . . . . . . . . . . . . 15  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) )
8786adantr 465 . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . 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 434 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 432 . . . . . . . . . 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 20 . . . . . . . . . . . . . . . 16  |-  ( f ( V SPaths  E ) p  ->  f ( V Walks  E ) p )
93923ad2ant1 1009 . . . . . . . . . . . . . . 15  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  f ( V Walks 
E ) p )
9493ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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 2445 . . . . . . . . . . . . . . . . . 18  |-  ( A  =  ( p ` 
0 )  <->  ( p `  0 )  =  A )
9695biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
97963ad2ant1 1009 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
98973ad2ant3 1011 . . . . . . . . . . . . . . 15  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
9998ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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 2445 . . . . . . . . . . . . . . . . . . . 20  |-  ( C  =  ( p ` 
2 )  <->  ( p `  2 )  =  C )
101100biimpi 194 . . . . . . . . . . . . . . . . . . 19  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
1021013ad2ant3 1011 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
103102, 48syl5ibr 221 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  f )  =  2  ->  (
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p `  ( # `  f
) )  =  C ) )
104103a1i 11 . . . . . . . . . . . . . . . 16  |-  ( f ( V SPaths  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p `  ( # `  f ) )  =  C ) ) )
1051043imp 1181 . . . . . . . . . . . . . . 15  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
106105ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 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 )
10794, 99, 1063jca 1168 . . . . . . . . . . . . 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  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C ) )
10846adantr 465 . . . . . . . . . . . . . 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 ) ) )
109108adantr 465 . . . . . . . . . . . . 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 ) ) )
110107, 109mpbird 232 . . . . . . . . . . . 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 )
111 id 22 . . . . . . . . . . . . . 14  |-  ( f ( V SPaths  E ) p  ->  f ( V SPaths  E ) p )
1121113ad2ant1 1009 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  f ( V SPaths  E ) p )
113112ad2antll 728 . . . . . . . . . . . 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 )
114110, 113jca 532 . . . . . . . . . . 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 ) )
115 id 22 . . . . . . . . . . . . 13  |-  ( (
# `  f )  =  2  ->  ( # `
 f )  =  2 )
1161153ad2ant2 1010 . . . . . . . . . . . 12  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( # `  f
)  =  2 )
117116ad2antll 728 . . . . . . . . . . 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 )
11822adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
119118adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  A  e.  V )
120 simpr 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  b  e.  V )
12133adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
122121adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  C  e.  V )
123119, 120, 1223jca 1168 . . . . . . . . . . . . . . . . . 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 )
)
124 oteqimp 30134 . . . . . . . . . . . . . . . . . . 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 ) ) )
125124imp 429 . . . . . . . . . . . . . . . . . 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 ) )
126123, 125sylan2 474 . . . . . . . . . . . . . . . . 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 ) )
127126ex 434 . . . . . . . . . . . . . . . 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 ) ) )
128 eqeq2 2452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  1 )  =  b  ->  (
( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  b ) )
129128eqcoms 2446 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  b ) )
1301293anbi2d 1294 . . . . . . . . . . . . . . . . 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 ) ) )
131130imbi2d 316 . . . . . . . . . . . . . . . 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 ) ) ) )
132127, 131syl5ibr 221 . . . . . . . . . . . . . . 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 ) ) ) )
1331323ad2ant2 1010 . . . . . . . . . . . . . 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 ) ) ) )
1341333ad2ant3 1011 . . . . . . . . . . . . 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 ) ) ) )
135134impcom 430 . . . . . . . . . . . 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 ) ) )
136135impcom 430 . . . . . . . . . . 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 ) )
137114, 117, 1363jca 1168 . . . . . . . . . 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 ) ) )
13891, 137jca 532 . . . . . . . . 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 ) ) ) )
139138ex 434 . . . . . . . 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 ) ) ) ) )
140139rexlimdva 2846 . . . . . . 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 ) ) ) ) )
14179, 140impbid 191 . . . . . 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 ) ) ) ) ) )
14221, 141bitrd 253 . . . . 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 ) ) ) ) ) )
1431422exbidv 1682 . . . 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 ) ) ) ) ) )
144 19.42vv 1926 . . . 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 ) ) ) )
145 rexcom4 2997 . . . . 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 ) ) ) ) )
146 rexcom4 2997 . . . . . 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 ) ) ) ) )
147146exbii 1634 . . . . 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 ) ) ) ) )
148145, 147bitr2i 250 . . . 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 ) ) ) ) )
149143, 144, 1483bitr3g 287 . . 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 ) ) ) ) ) )
150 19.42vv 1926 . . . 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
) ) ) ) )
151150rexbii 2745 . . 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 ) ) ) ) )
152149, 151syl6bb 261 . 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
) ) ) ) ) )
15314, 152bitrd 253 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 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2721   {crab 2724   _Vcvv 2977   <.cotp 3890   class class class wbr 4297    X. cxp 4843   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   0cc0 9287   1c1 9288   2c2 10376   #chash 12108   Walks cwalk 23410   Trails ctrail 23411   Paths cpath 23412   SPaths cspath 23413   WalkOn cwlkon 23414   SPathOn cspthon 23417   2SPathOnOt c2pthonot 30381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-ot 3891  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-wlk 23420  df-trail 23421  df-pth 23422  df-spth 23423  df-wlkon 23426  df-spthon 23429  df-2spthonot 30384
This theorem is referenced by:  el2spthonot0  30395  el2pthsot  30405  2spontn0vne  30411  2spotiundisj  30660
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