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Theorem el2wlkonot 25609
Description: A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2wlkonot  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V Walks  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 el2wlkonot
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 2wlkonot 25605 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( A ( V 2WalksOnOt  E ) C )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V WalkOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } )
21eleq2d 2516 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2WalksOnOt  E ) C )  <->  T  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V WalkOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } ) )
3 fveq2 5870 . . . . . . . . 9  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
43fveq2d 5874 . . . . . . . 8  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
54eqeq1d 2455 . . . . . . 7  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  T
) )  =  A ) )
63fveq2d 5874 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
76eqeq1d 2455 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
8 fveq2 5870 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
98eqeq1d 2455 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd `  T )  =  C ) )
105, 7, 93anbi123d 1341 . . . . . 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 1347 . . . . 5  |-  ( t  =  T  ->  (
( f ( A ( V WalkOn  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
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
12112exbidv 1772 . . . 4  |-  ( t  =  T  ->  ( E. f E. p ( f ( A ( V WalkOn  E ) C ) p  /\  ( # `
 f )  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) )  <->  E. f E. p
( f ( A ( V WalkOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
1312elrab 3198 . . 3  |-  ( T  e.  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V WalkOn  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 WalkOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
142, 13syl6bb 265 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2WalksOnOt  E ) C )  <->  ( T  e.  ( ( V  X.  V )  X.  V
)  /\  E. f E. p ( f ( A ( V WalkOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) ) )
15 simpl 459 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( V  e.  X  /\  E  e.  Y
) )
16 vex 3050 . . . . . . . . . . 11  |-  f  e. 
_V
17 vex 3050 . . . . . . . . . . 11  |-  p  e. 
_V
1816, 17pm3.2i 457 . . . . . . . . . 10  |-  ( f  e.  _V  /\  p  e.  _V )
1918a1i 11 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( f  e.  _V  /\  p  e.  _V )
)
20 simpr 463 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( A  e.  V  /\  C  e.  V
) )
21 iswlkon 25274 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
2215, 19, 20, 21syl3anc 1269 . . . . . . . 8  |-  ( ( ( 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 ) ) )
23223anbi1d 1345 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( f ( A ( V WalkOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) )  <->  ( ( f ( V Walks  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) )
2423anbi2d 711 . . . . . 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 )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  <-> 
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( f ( V Walks  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) ) )
25 simpl 459 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
2625adantl 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
2726adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  A  e.  V )
2827adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  /\  ( # `
 f )  =  2 )  ->  A  e.  V )
29 el2wlkonotlem 25602 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
3029ex 436 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
31303ad2ant1 1030 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
3231adantr 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  (
( # `  f )  =  2  ->  (
p `  1 )  e.  V ) )
3332imp 431 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
34 simpr 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3534adantl 468 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
3635adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  C  e.  V )
3736adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  /\  ( # `
 f )  =  2 )  ->  C  e.  V )
38 el2xptp0 6842 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 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 >. ) )
3928, 33, 37, 38syl3anc 1269 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  /\  ( # `
 f )  =  2 )  ->  (
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) )  <->  T  =  <. A ,  ( p `
 1 ) ,  C >. ) )
40 oteq2 4179 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( p `  1 )  =  b  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
4140eqcoms 2461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( b  =  ( p ` 
1 )  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
4241eqeq2d 2463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  <->  T  =  <. A ,  b ,  C >. )
)
4342biimpcd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( T  =  <. A ,  ( p `  1 ) ,  C >.  ->  (
b  =  ( p `
 1 )  ->  T  =  <. A , 
b ,  C >. ) )
4443adantr 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  ( b  =  ( p `  1
)  ->  T  =  <. A ,  b ,  C >. ) )
4544ad2antrr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  ( b  =  ( p `  1
)  ->  T  =  <. A ,  b ,  C >. ) )
4645impcom 432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  T  =  <. A ,  b ,  C >. )
47 simpl 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  f
( V Walks  E )
p )
4847adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  f ( V Walks 
E ) p )
4948ad2antrr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  f ( V Walks 
E ) p )
5049adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  f ( V Walks  E ) p )
51 simpr 463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  ( # `
 f )  =  2 )
5251adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  ( # `  f
)  =  2 )
5352ad2antrr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  ( # `  f
)  =  2 )
5453adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  ( # `  f
)  =  2 )
55 eqcom 2460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( p `  0 )  =  A  <->  A  =  ( p `  0
) )
5655biimpi 198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
5756adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  A  =  ( p `  0 ) )
5857adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  A  =  ( p `  0
) )
59 simpl 459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  b  =  ( p `  1
) )
60 fveq2 5870 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
6160eqeq1d 2455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
62 eqcom 2460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( p `  2 )  =  C  <->  C  =  ( p `  2
) )
6362biimpi 198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
6461, 63syl6bi 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  ->  C  =  ( p ` 
2 ) ) )
6564adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
( p `  ( # `
 f ) )  =  C  ->  C  =  ( p ` 
2 ) ) )
6665adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  ( ( p `
 ( # `  f
) )  =  C  ->  C  =  ( p `  2 ) ) )
6766imp 431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( T  =  <. A ,  ( p ` 
1 ) ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2 ) )  /\  ( p `
 ( # `  f
) )  =  C )  ->  C  =  ( p `  2
) )
6867adantr 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  C  =  ( p `  2 ) )
6968adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  C  =  ( p `  2
) )
7058, 59, 693jca 1189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )
7150, 54, 703jca 1189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  ( f
( V Walks  E )
p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
7246, 71jca 535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( b  =  ( p `
 1 )  /\  ( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A ) )  ->  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
7372ex 436 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( b  =  ( p ` 
1 )  ->  (
( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
7473adantl 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  /\  b  =  ( p `  1 ) )  ->  ( ( ( ( T  =  <. A ,  ( p ` 
1 ) ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2 ) )  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( p `
 0 )  =  A )  ->  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
7529, 74rspcimedv 3154 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
( ( ( T  =  <. A ,  ( p `  1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
7675com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
7776exp41 615 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( T  =  <. A ,  ( p `  1 ) ,  C >.  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  -> 
( ( p `  ( # `  f ) )  =  C  -> 
( ( p ` 
0 )  =  A  ->  ( ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) ) ) ) )
7877com15 96 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 )  -> 
( ( p `  ( # `  f ) )  =  C  -> 
( ( p ` 
0 )  =  A  ->  ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) ) ) ) )
7978pm2.43i 49 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
( p `  ( # `
 f ) )  =  C  ->  (
( p `  0
)  =  A  -> 
( T  =  <. A ,  ( p ` 
1 ) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) )
8079ex 436 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( (
p `  ( # `  f
) )  =  C  ->  ( ( p `
 0 )  =  A  ->  ( T  =  <. A ,  ( p `  1 ) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) ) ) ) )
8180com24 90 . . . . . . . . . . . . . . . . . . . . 21  |-  ( f ( V Walks  E ) p  ->  ( (
p `  0 )  =  A  ->  ( ( p `  ( # `  f ) )  =  C  ->  ( ( # `
 f )  =  2  ->  ( T  =  <. A ,  ( p `  1 ) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) ) ) ) )
82813imp 1203 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( ( # `
 f )  =  2  ->  ( T  =  <. A ,  ( p `  1 ) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) ) )
8382adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  (
( # `  f )  =  2  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) )
8483imp 431 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  /\  ( # `
 f )  =  2 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
8539, 84sylbid 219 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  /\  ( # `
 f )  =  2 )  ->  (
( 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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
8685ex 436 . . . . . . . . . . . . . . . 16  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
) )  ->  (
( # `  f )  =  2  ->  (
( 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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) )
8786com23 81 . . . . . . . . . . . . . . 15  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( ( 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 ) )  -> 
( ( # `  f
)  =  2  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) )
8887ex 436 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( (
( 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 ) )  -> 
( ( # `  f
)  =  2  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) )
8988com4t 88 . . . . . . . . . . . . 13  |-  ( ( T  e.  ( ( V  X.  V )  X.  V )  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) )  ->  ( ( # `
 f )  =  2  ->  ( (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( (
( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) )
9089ex 436 . . . . . . . . . . . 12  |-  ( T  e.  ( ( V  X.  V )  X.  V )  ->  (
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C )  ->  ( ( # `  f )  =  2  ->  ( ( f ( V Walks  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  ->  ( (
( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) ) )
9190com14 91 . . . . . . . . . . 11  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( (
( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C )  ->  ( ( # `  f )  =  2  ->  ( 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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) ) )
9291com23 81 . . . . . . . . . 10  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( ( # `
 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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) ) ) )
93923imp 1203 . . . . . . . . 9  |-  ( ( ( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( # `  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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) ) )
9493impcom 432 . . . . . . . 8  |-  ( ( T  e.  ( ( V  X.  V )  X.  V )  /\  ( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( # `  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 Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
9594com12 32 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  ->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
9625adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
97 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
9834adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
9996, 97, 983jca 1189 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
100 otel3xp 4873 . . . . . . . . . . . . . . . 16  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V
) )  ->  T  e.  ( ( V  X.  V )  X.  V
) )
10199, 100sylan2 477 . . . . . . . . . . . . . . 15  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) )
102101ex 436 . . . . . . . . . . . . . 14  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) )
103102adantr 467 . . . . . . . . . . . . 13  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
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
) ) )
104103com12 32 . . . . . . . . . . . 12  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) ) )
105104adantll 721 . . . . . . . . . . 11  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) ) )
106105imp 431 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) )
107 id 22 . . . . . . . . . . . . . . 15  |-  ( f ( V Walks  E ) p  ->  f ( V Walks  E ) p )
1081073ad2ant1 1030 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  f ( V Walks 
E ) p )
109108adantl 468 . . . . . . . . . . . . 13  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  f
( V Walks  E )
p )
110109adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
f ( V Walks  E
) p )
111 eqcom 2460 . . . . . . . . . . . . . . . . 17  |-  ( A  =  ( p ` 
0 )  <->  ( p `  0 )  =  A )
112111biimpi 198 . . . . . . . . . . . . . . . 16  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
1131123ad2ant1 1030 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
1141133ad2ant3 1032 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
115114adantl 468 . . . . . . . . . . . . 13  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  (
p `  0 )  =  A )
116115adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( p `  0
)  =  A )
117 eqcom 2460 . . . . . . . . . . . . . . . . . . 19  |-  ( C  =  ( p ` 
2 )  <->  ( p `  2 )  =  C )
118117biimpi 198 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
1191183ad2ant3 1032 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
120119, 61syl5ibr 225 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p `  ( # `  f
) )  =  C ) )
121120a1i 11 . . . . . . . . . . . . . . 15  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p `  ( # `  f ) )  =  C ) ) )
1221213imp 1203 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
123122adantl 468 . . . . . . . . . . . . 13  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  (
p `  ( # `  f
) )  =  C )
124123adantl 468 . . . . . . . . . . . 12  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( p `  ( # `
 f ) )  =  C )
125110, 116, 1243jca 1189 . . . . . . . . . . 11  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( f ( V Walks 
E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C ) )
126 id 22 . . . . . . . . . . . . . 14  |-  ( (
# `  f )  =  2  ->  ( # `
 f )  =  2 )
1271263ad2ant2 1031 . . . . . . . . . . . . 13  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( # `  f
)  =  2 )
128127adantl 468 . . . . . . . . . . . 12  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  ( # `
 f )  =  2 )
129128adantl 468 . . . . . . . . . . 11  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( # `  f )  =  2 )
13099adantll 721 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
131 oteqimp 6817 . . . . . . . . . . . . . . . . 17  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( A  e.  V  /\  b  e.  V  /\  C  e.  V
)  ->  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  b  /\  ( 2nd `  T )  =  C ) ) )
132130, 131syl5 33 . . . . . . . . . . . . . . . 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 ) ) )
133 eqeq2 2464 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  T ) )  =  b  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
1341333anbi2d 1346 . . . . . . . . . . . . . . . . 17  |-  ( b  =  ( p ` 
1 )  ->  (
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  b  /\  ( 2nd `  T
)  =  C )  <-> 
( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )
135134imbi2d 318 . . . . . . . . . . . . . . . 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
) )  =  b  /\  ( 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
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
136132, 135syl5ib 223 . . . . . . . . . . . . . . 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 ) ) ) )
1371363ad2ant2 1031 . . . . . . . . . . . . . 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 ) ) ) )
1381373ad2ant3 1032 . . . . . . . . . . . . 13  |-  ( ( f ( V Walks  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 ) ) ) )
139138impcom 432 . . . . . . . . . . . 12  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
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 ) ) )
140139impcom 432 . . . . . . . . . . 11  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
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 ) )
141125, 129, 1403jca 1189 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( ( f ( V Walks  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) )
142106, 141jca 535 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  /\  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )  -> 
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( f ( V Walks  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) )
143142ex 436 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  -> 
( T  e.  ( ( V  X.  V
)  X.  V )  /\  ( ( f ( V Walks  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) ) )
144143rexlimdva 2881 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( E. b  e.  V  ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) ) ) )
14595, 144impbid 194 . . . . . 6  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( ( T  e.  ( ( V  X.  V )  X.  V
)  /\  ( (
f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
14624, 145bitrd 257 . . . . 5  |-  ( ( ( 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 )  =  2  /\  ( ( 1st `  ( 1st `  T
) )  =  A  /\  ( 2nd `  ( 1st `  T ) )  =  ( p ` 
1 )  /\  ( 2nd `  T )  =  C ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
1471462exbidv 1772 . . . 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 WalkOn  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 Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) ) )
148 19.42vv 1838 . . . 4  |-  ( E. f E. p ( T  e.  ( ( V  X.  V )  X.  V )  /\  ( f ( A ( V WalkOn  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 WalkOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 )  /\  ( 2nd `  T
)  =  C ) ) ) )
149 rexcom4 3069 . . . . 5  |-  ( E. b  e.  V  E. f E. p ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
150 rexcom4 3069 . . . . . 6  |-  ( E. b  e.  V  E. p ( T  = 
<. A ,  b ,  C >.  /\  (
f ( V Walks  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 Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
151150exbii 1720 . . . . 5  |-  ( E. f E. b  e.  V  E. p ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
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 Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
152149, 151bitr2i 254 . . . 4  |-  ( E. f E. p E. b  e.  V  ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
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 Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) )
153147, 148, 1523bitr3g 291 . . 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 WalkOn  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 Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) ) ) )
154 19.42vv 1838 . . . 4  |-  ( E. f E. p ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) )
155154rexbii 2891 . . 3  |-  ( E. b  e.  V  E. f E. p ( T  =  <. A ,  b ,  C >.  /\  (
f ( V Walks  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 Walks  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
156153, 155syl6bb 265 . 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 WalkOn  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 Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
15714, 156bitrd 257 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446   E.wex 1665    e. wcel 1889   E.wrex 2740   {crab 2743   _Vcvv 3047   <.cotp 3978   class class class wbr 4405    X. cxp 4835   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797   0cc0 9544   1c1 9545   2c2 10666   #chash 12522   Walks cwalk 25238   WalkOn cwlkon 25242   2WalksOnOt c2wlkonot 25595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-ot 3979  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-wlk 25248  df-wlkon 25254  df-2wlkonot 25598
This theorem is referenced by:  el2wlkonotot0  25612  el2wlksot  25620  frg2wot1  25797  frg2woteq  25800
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