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Theorem el2wlkonot 28066
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 28062 . . . 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 2471 . . 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 5687 . . . . . . . . 9  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
43fveq2d 5691 . . . . . . . 8  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
54eqeq1d 2412 . . . . . . 7  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  T
) )  =  A ) )
63fveq2d 5691 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
76eqeq1d 2412 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
8 fveq2 5687 . . . . . . . 8  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
98eqeq1d 2412 . . . . . . 7  |-  ( t  =  T  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd `  T )  =  C ) )
105, 7, 93anbi123d 1254 . . . . . 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 1260 . . . . 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 1635 . . . 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 3052 . . 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 253 . 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 444 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( V  e.  X  /\  E  e.  Y
) )
16 vex 2919 . . . . . . . . . . 11  |-  f  e. 
_V
17 vex 2919 . . . . . . . . . . 11  |-  p  e. 
_V
1816, 17pm3.2i 442 . . . . . . . . . 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 448 . . . . . . . . 9  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( A  e.  V  /\  C  e.  V
) )
21 iswlkon 21484 . . . . . . . . 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 1184 . . . . . . . 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 1258 . . . . . . 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 685 . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
2625adantl 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
2726adantl 453 . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 28059 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
p `  1 )  e.  V )
3029ex 424 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( f ( V Walks  E ) p  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
31303ad2ant1 978 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f ( V Walks  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( ( # `
 f )  =  2  ->  ( p `  1 )  e.  V ) )
3231adantr 452 . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3534adantl 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
3635adantl 453 . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 27949 . . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . . 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 3954 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( p `  1 )  =  b  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
4140eqcoms 2407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( b  =  ( p ` 
1 )  ->  <. A , 
( p `  1
) ,  C >.  = 
<. A ,  b ,  C >. )
4241eqeq2d 2415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( b  =  ( p ` 
1 )  ->  ( T  =  <. A , 
( p `  1
) ,  C >.  <->  T  =  <. A ,  b ,  C >. )
)
4342biimpcd 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( T  =  <. A ,  ( p `  1 ) ,  C >.  ->  (
b  =  ( p `
 1 )  ->  T  =  <. A , 
b ,  C >. ) )
4443adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  ( b  =  ( p `  1
)  ->  T  =  <. A ,  b ,  C >. ) )
4544ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  f
( V Walks  E )
p )
4847adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  f ( V Walks 
E ) p )
4948ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  ( # `
 f )  =  2 )
5251adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  ( # `  f
)  =  2 )
5352ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  ( # `  f
)  =  2 )
5453adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( p `  0 )  =  A  <->  A  =  ( p `  0
) )
5655biimpi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
5756adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  A  =  ( p `  0 ) )
5857adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
6160eqeq1d 2412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
62 eqcom 2406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40  |-  ( ( p `  2 )  =  C  <->  C  =  ( p `  2
) )
6362biimpi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
6461, 63syl6bi 220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  ->  C  =  ( p ` 
2 ) ) )
6564adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2 )  ->  (
( p `  ( # `
 f ) )  =  C  ->  C  =  ( p ` 
2 ) ) )
6665adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( ( T  =  <. A , 
( p `  1
) ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2 ) )  ->  ( ( p `
 ( # `  f
) )  =  C  ->  C  =  ( p `  2 ) ) )
6766imp 419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( T  =  <. A ,  ( p ` 
1 ) ,  C >.  /\  ( f ( V Walks  E ) p  /\  ( # `  f
)  =  2 ) )  /\  ( p `
 ( # `  f
) )  =  C )  ->  C  =  ( p `  2
) )
6867adantr 452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( T  = 
<. A ,  ( p `
 1 ) ,  C >.  /\  (
f ( V Walks  E
) p  /\  ( # `
 f )  =  2 ) )  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( p `  0
)  =  A )  ->  C  =  ( p `  2 ) )
6968adantl 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3014 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29 . . . . . . . . . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . . . . . . . . . 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 89 . . . . . . . . . . . . . . . . . . . . . . . 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 45 . . . . . . . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . . . . . . . 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 83 . . . . . . . . . . . . . . . . . . . . 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 1147 . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . . . 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 74 . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . 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 81 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 84 . . . . . . . . . . 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 74 . . . . . . . . . 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 1147 . . . . . . . . 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 420 . . . . . . . 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 29 . . . . . . 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 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
97 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
9834adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
9996, 97, 983jca 1134 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  ( A  e.  V  /\  b  e.  V  /\  C  e.  V )
)
100 otel3xp 27950 . . . . . . . . . . . . . . . 16  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V
) )  ->  T  e.  ( ( V  X.  V )  X.  V
) )
10199, 100sylan2 461 . . . . . . . . . . . . . . 15  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V ) )  ->  T  e.  ( ( V  X.  V )  X.  V ) )
102101ex 424 . . . . . . . . . . . . . 14  |-  ( T  =  <. A ,  b ,  C >.  ->  (
( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  ->  T  e.  ( ( V  X.  V )  X.  V
) ) )
103102adantr 452 . . . . . . . . . . . . 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 29 . . . . . . . . . . . 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 695 . . . . . . . . . . 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 419 . . . . . . . . . 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 20 . . . . . . . . . . . . . . 15  |-  ( f ( V Walks  E ) p  ->  f ( V Walks  E ) p )
1081073ad2ant1 978 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  f ( V Walks 
E ) p )
109108adantl 453 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 2406 . . . . . . . . . . . . . . . . 17  |-  ( A  =  ( p ` 
0 )  <->  ( p `  0 )  =  A )
112111biimpi 187 . . . . . . . . . . . . . . . 16  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
1131123ad2ant1 978 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
1141133ad2ant3 980 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
115114adantl 453 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 2406 . . . . . . . . . . . . . . . . . . 19  |-  ( C  =  ( p ` 
2 )  <->  ( p `  2 )  =  C )
118117biimpi 187 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
1191183ad2ant3 980 . . . . . . . . . . . . . . . . 17  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
120119, 61syl5ibr 213 . . . . . . . . . . . . . . . 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 1147 . . . . . . . . . . . . . 14  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
123122adantl 453 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 1134 . . . . . . . . . . 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 20 . . . . . . . . . . . . . 14  |-  ( (
# `  f )  =  2  ->  ( # `
 f )  =  2 )
1271263ad2ant2 979 . . . . . . . . . . . . 13  |-  ( ( f ( V Walks  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( # `  f
)  =  2 )
128127adantl 453 . . . . . . . . . . . 12  |-  ( ( T  =  <. A , 
b ,  C >.  /\  ( f ( V Walks 
E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  ->  ( # `
 f )  =  2 )
129128adantl 453 . . . . . . . . . . 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 695 . . . . . . . . . . . . . . . . 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 27951 . . . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . . . 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 2413 . . . . . . . . . . . . . . . . . 18  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  T ) )  =  b  <->  ( 2nd `  ( 1st `  T
) )  =  ( p `  1 ) ) )
1341333anbi2d 1259 . . . . . . . . . . . . . . . . 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 308 . . . . . . . . . . . . . . . 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 211 . . . . . . . . . . . . . . 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 979 . . . . . . . . . . . . . 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 980 . . . . . . . . . . . . 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 420 . . . . . . . . . . . 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 420 . . . . . . . . . . 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 1134 . . . . . . . . . 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 519 . . . . . . . . 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 424 . . . . . . . 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 2790 . . . . . . 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 184 . . . . . 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 245 . . . . 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 1635 . . . 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 1926 . . . 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 2935 . . . . 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 2935 . . . . . 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 1589 . . . . 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 242 . . . 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 279 . . 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 1926 . . . 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 2691 . . 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 253 . 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 245 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916   <.cotp 3778   class class class wbr 4172    X. cxp 4835   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   0cc0 8946   1c1 8947   2c2 10005   #chash 11573   Walks cwalk 21459   WalkOn cwlkon 21463   2WalksOnOt c2wlkonot 28052
This theorem is referenced by:  el2wlkonotot0  28069  el2wlksot  28077  frg2wot1  28160  frg2woteq  28163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-word 11678  df-wlk 21469  df-wlkon 21475  df-2wlkonot 28055
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