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Theorem el2spthonot0 28403
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 28402 . 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 5731 . . . . . . . . . . . 12  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  t )  =  ( 1st `  <. A ,  b ,  C >. ) )
32fveq2d 5735 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  <. A ,  b ,  C >. )
) )
43eqeq1d 2446 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A ) )
52fveq2d 5735 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
) )
65eqeq1d 2446 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
7 fveq2 5731 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  t )  =  ( 2nd `  <. A ,  b ,  C >. ) )
87eqeq1d 2446 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd ` 
<. A ,  b ,  C >. )  =  C ) )
94, 6, 83anbi123d 1255 . . . . . . . . 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 1261 . . . . . . . 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 1639 . . . . . . 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 3094 . . . . . 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 28398 . . . . . . 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 453 . . . . . 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 2505 . . . . 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 964 . . . . . . . . . . . 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 21 . . . . . . . . . . . . . . . 16  |-  ( ( p `  0 )  =  A  ->  (
p `  0 )  =  A )
1918eqcoms 2441 . . . . . . . . . . . . . . 15  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
20193ad2ant1 979 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
21203ad2ant3 981 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
2221adantl 454 . . . . . . . . . . . 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 5731 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
24 id 21 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  2 )  =  C  ->  (
p `  2 )  =  C )
2524eqcoms 2441 . . . . . . . . . . . . . . . 16  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
26253ad2ant3 981 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
2723, 26sylan9eq 2490 . . . . . . . . . . . . . 14  |-  ( ( ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  (
p `  ( # `  f
) )  =  C )
28273adant1 976 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
2928adantl 454 . . . . . . . . . . . 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 1135 . . . . . . . . . . 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 965 . . . . . . . . . . 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 2439 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
33 simpl 445 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
3433adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
35 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
36 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3736adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
38 oteqimp 28078 . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . 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 1187 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 2447 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
43423ad2ant2 980 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
44433ad2ant3 981 . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . 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 1260 . . . . . . . . . . . 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 203 . . . . . . . . . . 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 1135 . . . . . . . . . 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 1042 . . . . . . . . . . 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 965 . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
52513anbi3d 1261 . . . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 0 )  e. 
_V
54 eleq1 2498 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  (
( p `  0
)  e.  _V  <->  A  e.  _V ) )
5553, 54mpbii 204 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  0 )  =  A  ->  A  e.  _V )
5655adantr 453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  e.  _V )
57 vex 2961 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  b  e. 
_V
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  b  e.  _V )
59 fvex 5745 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 2 )  e. 
_V
60 eleq1 2498 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  2 )  =  C  ->  (
( p `  2
)  e.  _V  <->  C  e.  _V ) )
6159, 60mpbii 204 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  e.  _V )
6261adantl 454 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  C  e.  _V )
63 ot2ndg 6365 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  e.  _V  /\  b  e.  _V  /\  C  e.  _V )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b )
6456, 58, 62, 63syl3anc 1185 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b )
6564eqeq1d 2446 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 )  <-> 
b  =  ( p `
 1 ) ) )
66 id 21 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( A  =  ( p ` 
0 )  ->  A  =  ( p ` 
0 ) )
6766eqcoms 2441 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
6867adantr 453 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  =  ( p `  0 ) )
6968adantr 453 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  A  =  ( p ` 
0 ) )
70 simpr 449 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  b  =  ( p ` 
1 ) )
71 id 21 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( C  =  ( p ` 
2 )  ->  C  =  ( p ` 
2 ) )
7271eqcoms 2441 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
7372ad2antlr 709 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  C  =  ( p ` 
2 ) )
7469, 70, 733jca 1135 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )
7574ex 425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
7665, 75sylbid 208 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )
7776com12 30 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  -> 
( ( ( p `
 0 )  =  A  /\  ( p `
 2 )  =  C )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
78773ad2ant2 980 . . . . . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . . . . . 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 976 . . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . 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 30 . . . . . . . . . . . . 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 1148 . . . . . . . . . . . 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 454 . . . . . . . . . . 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 1135 . . . . . . . . . 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 807 . . . . . . . . 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 696 . . . . . . . 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 2961 . . . . . . . . . . . . 13  |-  f  e. 
_V
90 vex 2961 . . . . . . . . . . . . 13  |-  p  e. 
_V
9189, 90pm3.2i 443 . . . . . . . . . . . 12  |-  ( f  e.  _V  /\  p  e.  _V )
92 isspthonpth 21589 . . . . . . . . . . . 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 1268 . . . . . . . . . . 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 453 . . . . . . . . . 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 194 . . . . . . . . 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 1259 . . . . . . . 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 246 . . . . . . 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 1639 . . . . . 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 2439 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
10033ad2antlr 709 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  A  e.  V )
101 simpr 449 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  b  e.  V )
10236ad2antlr 709 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  C  e.  V )
103 otel3xp 28077 . . . . . . . 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 1187 . . . . . . 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 496 . . . . . 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 246 . . . . 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 279 . . . 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 686 . . 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 2724 . 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 246 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   {crab 2711   _Vcvv 2958   <.cotp 3820   class class class wbr 4215    X. cxp 4879   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   0cc0 8995   1c1 8996   2c2 10054   #chash 11623   SPaths cspath 21514   SPathOn cspthon 21518   2SPathOnOt c2pthonot 28389
This theorem is referenced by:  usg2spthonot1  28422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-fzo 11141  df-hash 11624  df-word 11728  df-wlk 21521  df-trail 21522  df-pth 21523  df-spth 21524  df-wlkon 21527  df-spthon 21530  df-2spthonot 28392
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