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Theorem el2spthonot 24574
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 24570 . . . 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 2537 . . 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 5866 . . . . . . . . 9  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
43fveq2d 5870 . . . . . . . 8  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
54eqeq1d 2469 . . . . . . 7  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  T
) )  =  A ) )
63fveq2d 5870 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
76eqeq1d 2469 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
8 fveq2 5866 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
98eqeq1d 2469 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd `  T )  =  C ) )
105, 7, 93anbi123d 1299 . . . . . 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 1305 . . . . 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 1692 . . . 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 3261 . . 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 3116 . . . . . . . . . 10  |-  f  e. 
_V
16 vex 3116 . . . . . . . . . 10  |-  p  e. 
_V
1715, 16pm3.2i 455 . . . . . . . . 9  |-  ( f  e.  _V  /\  p  e.  _V )
18 isspthon 24289 . . . . . . . . 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 1312 . . . . . . . 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 1303 . . . . . . 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 24279 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V SPaths  E ) p  ->  f ( V Paths  E ) p )
25 pthistrl 24278 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Paths  E ) p  ->  f ( V Trails  E ) p )
26 trliswlk 24245 . . . . . . . . . . . . . . . . . . 19  |-  ( f ( V Trails  E ) p  ->  f ( V Walks  E ) p )
27 el2wlkonotlem 24566 . . . . . . . . . . . . . . . . . . . 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 6828 . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . 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 4223 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( p `  1 )  =  b  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
3837eqcoms 2479 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  ( p ` 
1 )  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
3938eqeq2d 2481 . . . . . . . . . . . . . . . . . 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 24238 . . . . . . . . . . . . . . . . . . . . . 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 1312 . . . . . . . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
4847eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . 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 3216 . . . . . . . . . . . . . 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 1190 . . . . . . . . 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 1176 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
84 otel3xp 5035 . . . . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . 18  |-  ( A  =  ( p ` 
0 )  <->  ( p `  0 )  =  A )
9695biimpi 194 . . . . . . . . . . . . . . . . 17  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
97963ad2ant1 1017 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
98973ad2ant3 1019 . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . . . . . . . 20  |-  ( C  =  ( p ` 
2 )  <->  ( p `  2 )  =  C )
101100biimpi 194 . . . . . . . . . . . . . . . . . . 19  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
1021013ad2ant3 1019 . . . . . . . . . . . . . . . . . 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 1190 . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . 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 1018 . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . . . 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 6803 . . . . . . . . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  1 )  =  b  ->  (
( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  b ) )
129128eqcoms 2479 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  b ) )
1301293anbi2d 1304 . . . . . . . . . . . . . . . . 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 1018 . . . . . . . . . . . . . 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 1019 . . . . . . . . . . . . 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 1176 . . . . . . . . . 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 2955 . . . . . . 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 1692 . . . 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 1951 . . . 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 3133 . . . . 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 3133 . . . . . 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 1644 . . . . 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 1951 . . . 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 2965 . . 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 973    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113   <.cotp 4035   class class class wbr 4447    X. cxp 4997   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   0cc0 9492   1c1 9493   2c2 10585   #chash 12373   Walks cwalk 24202   Trails ctrail 24203   Paths cpath 24204   SPaths cspath 24205   WalkOn cwlkon 24206   SPathOn cspthon 24209   2SPathOnOt c2pthonot 24561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-wlk 24212  df-trail 24213  df-pth 24214  df-spth 24215  df-wlkon 24218  df-spthon 24221  df-2spthonot 24564
This theorem is referenced by:  el2spthonot0  24575  el2pthsot  24585  2spontn0vne  24591  2spotiundisj  24767
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