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Theorem el2spthonot0 25655
Description: A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
Assertion
Ref Expression
el2spthonot0  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
Distinct variable groups:    A, b    C, b    E, b    T, b    V, b    X, b    Y, b

Proof of Theorem el2spthonot0
Dummy variables  f  p  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 el2spthonot 25654 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) ) ) )
2 fveq2 5892 . . . . . . . . . . . 12  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  t )  =  ( 1st `  <. A ,  b ,  C >. ) )
32fveq2d 5896 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  <. A ,  b ,  C >. )
) )
43eqeq1d 2464 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A ) )
52fveq2d 5896 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
) )
65eqeq1d 2464 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
7 fveq2 5892 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  t )  =  ( 2nd `  <. A ,  b ,  C >. ) )
87eqeq1d 2464 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd ` 
<. A ,  b ,  C >. )  =  C ) )
94, 6, 83anbi123d 1348 . . . . . . . . 9  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C )  <-> 
( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )
1093anbi3d 1354 . . . . . . . 8  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) )  <->  ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
11102exbidv 1781 . . . . . . 7  |-  ( t  =  <. A ,  b ,  C >.  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  C ) )  <->  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
1211elrab 3208 . . . . . 6  |-  ( <. A ,  b ,  C >.  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  C ) ) }  <-> 
( <. A ,  b ,  C >.  e.  ( ( V  X.  V
)  X.  V )  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
1312a1i 11 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( <. A ,  b ,  C >.  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  C ) ) }  <-> 
( <. A ,  b ,  C >.  e.  ( ( V  X.  V
)  X.  V )  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) ) )
14 2spthonot 25650 . . . . . . 7  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( A ( V 2SPathOnOt  E ) C )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } )
1514adantr 471 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( A ( V 2SPathOnOt  E ) C )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } )
1615eleq2d 2525 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <->  <. A , 
b ,  C >.  e. 
{ t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } ) )
17 simpr1 1020 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  f ( V SPaths  E ) p )
18 id 22 . . . . . . . . . . . . . . . 16  |-  ( ( p `  0 )  =  A  ->  (
p `  0 )  =  A )
1918eqcoms 2470 . . . . . . . . . . . . . . 15  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
20193ad2ant1 1035 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
21203ad2ant3 1037 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
2221adantl 472 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( p `  0 )  =  A )
23 fveq2 5892 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
24 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  2 )  =  C  ->  (
p `  2 )  =  C )
2524eqcoms 2470 . . . . . . . . . . . . . . . 16  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
26253ad2ant3 1037 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
2723, 26sylan9eq 2516 . . . . . . . . . . . . . 14  |-  ( ( ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  (
p `  ( # `  f
) )  =  C )
28273adant1 1032 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
2928adantl 472 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( p `  ( # `  f
) )  =  C )
3017, 22, 293jca 1194 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( f
( V SPaths  E )
p  /\  ( p `  0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C ) )
31 simpr2 1021 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( # `  f
)  =  2 )
32 eqidd 2463 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
33 simpl 463 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
3433adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
35 simpr 467 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
36 simpr 467 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3736adantr 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
38 oteqimp 6844 . . . . . . . . . . . . . . 15  |-  ( <. A ,  b ,  C >.  =  <. A , 
b ,  C >.  -> 
( ( A  e.  V  /\  b  e.  V  /\  C  e.  V )  ->  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )
3938imp 435 . . . . . . . . . . . . . 14  |-  ( (
<. A ,  b ,  C >.  =  <. A ,  b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V ) )  -> 
( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )
4032, 34, 35, 37, 39syl13anc 1278 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )
4140adantr 471 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )
42 eqeq2 2473 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
43423ad2ant2 1036 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
44433ad2ant3 1037 . . . . . . . . . . . . . 14  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
4544adantl 472 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
46453anbi2d 1353 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  <->  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )
4741, 46mpbid 215 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )
4830, 31, 473jca 1194 . . . . . . . . . 10  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )  ->  ( (
f ( V SPaths  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )
49 simpr11 1098 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )  ->  f ( V SPaths  E ) p )
50 simpr2 1021 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )  ->  ( # `  f
)  =  2 )
5123eqeq1d 2464 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
52513anbi3d 1354 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  <->  ( f
( V SPaths  E )
p  /\  ( p `  0 )  =  A  /\  ( p `
 2 )  =  C ) ) )
53 fvex 5902 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 0 )  e. 
_V
54 eleq1 2528 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  (
( p `  0
)  e.  _V  <->  A  e.  _V ) )
5553, 54mpbii 216 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  0 )  =  A  ->  A  e.  _V )
5655adantr 471 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  e.  _V )
57 vex 3060 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  b  e. 
_V
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  b  e.  _V )
59 fvex 5902 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 2 )  e. 
_V
60 eleq1 2528 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  2 )  =  C  ->  (
( p `  2
)  e.  _V  <->  C  e.  _V ) )
6159, 60mpbii 216 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  e.  _V )
6261adantl 472 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  C  e.  _V )
63 ot2ndg 6840 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  e.  _V  /\  b  e.  _V  /\  C  e.  _V )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b )
6456, 58, 62, 63syl3anc 1276 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b )
6564eqeq1d 2464 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 )  <-> 
b  =  ( p `
 1 ) ) )
66 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( A  =  ( p ` 
0 )  ->  A  =  ( p ` 
0 ) )
6766eqcoms 2470 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
6867adantr 471 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  =  ( p `  0 ) )
6968adantr 471 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  A  =  ( p ` 
0 ) )
70 simpr 467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  b  =  ( p ` 
1 ) )
71 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( C  =  ( p ` 
2 )  ->  C  =  ( p ` 
2 ) )
7271eqcoms 2470 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
7372ad2antlr 738 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  C  =  ( p ` 
2 ) )
7469, 70, 733jca 1194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )
7574ex 440 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
7665, 75sylbid 223 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )
7776com12 32 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  -> 
( ( ( p `
 0 )  =  A  /\  ( p `
 2 )  =  C )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
78773ad2ant2 1036 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  -> 
( ( ( p `
 0 )  =  A  /\  ( p `
 2 )  =  C )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
7978com12 32 . . . . . . . . . . . . . . . . 17  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  -> 
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
80793adant1 1032 . . . . . . . . . . . . . . . 16  |-  ( ( f ( V SPaths  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  2 )  =  C )  ->  (
( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  -> 
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
8180a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  -> 
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
8252, 81sylbid 223 . . . . . . . . . . . . . 14  |-  ( (
# `  f )  =  2  ->  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  -> 
( ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  -> 
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
8382com12 32 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  (
p `  0 )  =  A  /\  (
p `  ( # `  f
) )  =  C )  ->  ( ( # `
 f )  =  2  ->  ( (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C )  -> 
( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) ) )
84833imp 1208 . . . . . . . . . . . 12  |-  ( ( ( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) )
8584adantl 472 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )
8649, 50, 853jca 1194 . . . . . . . . . 10  |-  ( ( ( ( A  e.  V  /\  C  e.  V )  /\  b  e.  V )  /\  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) )  ->  ( f
( V SPaths  E )
p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
8748, 86impbida 848 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  (
( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  <->  ( ( f ( V SPaths  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
8887adantll 725 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  <->  ( ( f ( V SPaths  E ) p  /\  ( p `
 0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C )  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
89 vex 3060 . . . . . . . . . . . . 13  |-  f  e. 
_V
90 vex 3060 . . . . . . . . . . . . 13  |-  p  e. 
_V
9189, 90pm3.2i 461 . . . . . . . . . . . 12  |-  ( f  e.  _V  /\  p  e.  _V )
92 isspthonpth 25370 . . . . . . . . . . . 12  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( f  e.  _V  /\  p  e. 
_V )  /\  ( A  e.  V  /\  C  e.  V )
)  ->  ( f
( A ( V SPathOn  E ) C ) p  <->  ( f ( V SPaths  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C ) ) )
9391, 92mp3an2 1361 . . . . . . . . . . 11  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( f ( A ( V SPathOn  E ) C ) p  <->  ( f
( V SPaths  E )
p  /\  ( p `  0 )  =  A  /\  ( p `
 ( # `  f
) )  =  C ) ) )
9493adantr 471 . . . . . . . . . 10  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
f ( A ( V SPathOn  E ) C ) p  <->  ( f ( V SPaths  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C ) ) )
9594bicomd 206 . . . . . . . . 9  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( f ( V SPaths  E ) p  /\  ( p `  0
)  =  A  /\  ( p `  ( # `
 f ) )  =  C )  <->  f ( A ( V SPathOn  E
) C ) p ) )
96953anbi1d 1352 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( ( f ( V SPaths  E ) p  /\  ( p ` 
0 )  =  A  /\  ( p `  ( # `  f ) )  =  C )  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )  <-> 
( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
9788, 96bitrd 261 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  <->  ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
98972exbidv 1781 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( E. f E. p ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  <->  E. f E. p ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) )
99 eqidd 2463 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
10033ad2antlr 738 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  A  e.  V )
101 simpr 467 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  b  e.  V )
10236ad2antlr 738 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  C  e.  V )
103 otel3xp 4892 . . . . . . . 8  |-  ( (
<. A ,  b ,  C >.  =  <. A ,  b ,  C >.  /\  ( A  e.  V  /\  b  e.  V  /\  C  e.  V ) )  ->  <. A ,  b ,  C >.  e.  (
( V  X.  V
)  X.  V ) )
10499, 100, 101, 102, 103syl13anc 1278 . . . . . . 7  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  e.  ( ( V  X.  V )  X.  V
) )
105104biantrurd 515 . . . . . 6  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( E. f E. p ( f ( A ( V SPathOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) )  <-> 
( <. A ,  b ,  C >.  e.  ( ( V  X.  V
)  X.  V )  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) ) )
10698, 105bitrd 261 . . . . 5  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( E. f E. p ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  <-> 
( <. A ,  b ,  C >.  e.  ( ( V  X.  V
)  X.  V )  /\  E. f E. p ( f ( A ( V SPathOn  E
) C ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  <. A ,  b ,  C >. )
)  =  A  /\  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  /\  ( 2nd `  <. A , 
b ,  C >. )  =  C ) ) ) ) )
10713, 16, 1063bitr4rd 294 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  ( E. f E. p ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  <->  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) )
108107anbi2d 715 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  (
( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
109108rexbidva 2910 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( E. b  e.  V  ( T  = 
<. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
1101, 109bitrd 261 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   E.wrex 2750   {crab 2753   _Vcvv 3057   <.cotp 3988   class class class wbr 4418    X. cxp 4854   ` cfv 5605  (class class class)co 6320   1stc1st 6823   2ndc2nd 6824   0cc0 9570   1c1 9571   2c2 10692   #chash 12553   SPaths cspath 25285   SPathOn cspthon 25289   2SPathOnOt c2pthonot 25641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-ot 3989  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-pm 7506  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-card 8404  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-2 10701  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-fzo 11953  df-hash 12554  df-word 12703  df-wlk 25292  df-trail 25293  df-pth 25294  df-spth 25295  df-wlkon 25298  df-spthon 25301  df-2spthonot 25644
This theorem is referenced by:  usg2spthonot1  25674
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