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Theorem el2spthonot0 24685
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 24684 . 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 5872 . . . . . . . . . . . 12  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  t )  =  ( 1st `  <. A ,  b ,  C >. ) )
32fveq2d 5876 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  <. A ,  b ,  C >. )
) )
43eqeq1d 2469 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A ) )
52fveq2d 5876 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
) )
65eqeq1d 2469 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
7 fveq2 5872 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  t )  =  ( 2nd `  <. A ,  b ,  C >. ) )
87eqeq1d 2469 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd ` 
<. A ,  b ,  C >. )  =  C ) )
94, 6, 83anbi123d 1299 . . . . . . . . 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 1305 . . . . . . . 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 1692 . . . . . . 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 3266 . . . . . 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 24680 . . . . . . 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 465 . . . . . 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 2537 . . . . 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 1002 . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . 15  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
20193ad2ant1 1017 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
21203ad2ant3 1019 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p ` 
0 )  =  A )
2221adantl 466 . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
24 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  2 )  =  C  ->  (
p `  2 )  =  C )
2524eqcoms 2479 . . . . . . . . . . . . . . . 16  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
26253ad2ant3 1019 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
2723, 26sylan9eq 2528 . . . . . . . . . . . . . 14  |-  ( ( ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  (
p `  ( # `  f
) )  =  C )
28273adant1 1014 . . . . . . . . . . . . 13  |-  ( ( f ( V SPaths  E
) p  /\  ( # `
 f )  =  2  /\  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) )  ->  ( p `  ( # `  f ) )  =  C )
2928adantl 466 . . . . . . . . . . . 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 1176 . . . . . . . . . . 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 1003 . . . . . . . . . . 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 2468 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
33 simpl 457 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  C  e.  V )  ->  A  e.  V )
3433adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  A  e.  V )
35 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  b  e.  V )
36 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  V  /\  C  e.  V )  ->  C  e.  V )
3736adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  C  e.  V
)  /\  b  e.  V )  ->  C  e.  V )
38 oteqimp 6814 . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 2482 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
43423ad2ant2 1018 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
44433ad2ant3 1019 . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . 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 1304 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 1176 . . . . . . . . . 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 1080 . . . . . . . . . . 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 1003 . . . . . . . . . . 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 2469 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
52513anbi3d 1305 . . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 0 )  e. 
_V
54 eleq1 2539 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  (
( p `  0
)  e.  _V  <->  A  e.  _V ) )
5553, 54mpbii 211 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  0 )  =  A  ->  A  e.  _V )
5655adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  e.  _V )
57 vex 3121 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  b  e. 
_V
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  b  e.  _V )
59 fvex 5882 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 2 )  e. 
_V
60 eleq1 2539 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  2 )  =  C  ->  (
( p `  2
)  e.  _V  <->  C  e.  _V ) )
6159, 60mpbii 211 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  e.  _V )
6261adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  C  e.  _V )
63 ot2ndg 6810 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  e.  _V  /\  b  e.  _V  /\  C  e.  _V )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b )
6456, 58, 62, 63syl3anc 1228 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b )
6564eqeq1d 2469 . . . . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  A  =  ( p ` 
0 ) )
6867adantr 465 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  =  ( p `  0 ) )
6968adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  A  =  ( p ` 
0 ) )
70 simpr 461 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  b  =  ( p ` 
1 ) )
71 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( C  =  ( p ` 
2 )  ->  C  =  ( p ` 
2 ) )
7271eqcoms 2479 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
7372ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  C  =  ( p ` 
2 ) )
7469, 70, 733jca 1176 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )
7574ex 434 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( b  =  ( p `  1
)  ->  ( A  =  ( p ` 
0 )  /\  b  =  ( p ` 
1 )  /\  C  =  ( p ` 
2 ) ) ) )
7665, 75sylbid 215 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 )  ->  ( A  =  ( p `  0
)  /\  b  =  ( p `  1
)  /\  C  =  ( p `  2
) ) ) )
7776com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 )  -> 
( ( ( p `
 0 )  =  A  /\  ( p `
 2 )  =  C )  ->  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) ) )
78773ad2ant2 1018 . . . . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . 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 1190 . . . . . . . . . . . 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 466 . . . . . . . . . . 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 1176 . . . . . . . . . 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 830 . . . . . . . . 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 713 . . . . . . . 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 3121 . . . . . . . . . . . . 13  |-  f  e. 
_V
90 vex 3121 . . . . . . . . . . . . 13  |-  p  e. 
_V
9189, 90pm3.2i 455 . . . . . . . . . . . 12  |-  ( f  e.  _V  /\  p  e.  _V )
92 isspthonpth 24400 . . . . . . . . . . . 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 1312 . . . . . . . . . . 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 465 . . . . . . . . . 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 201 . . . . . . . . 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 1303 . . . . . . . 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 253 . . . . . . 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 1692 . . . . . 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 2468 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
10033ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  A  e.  V )
101 simpr 461 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  b  e.  V )
10236ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  C  e.  V )
103 otel3xp 5041 . . . . . . . 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 1230 . . . . . . 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 508 . . . . . 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 253 . . . . 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 286 . . . 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 703 . . 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 2975 . 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 253 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  C  e.  V ) )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2818   {crab 2821   _Vcvv 3118   <.cotp 4041   class class class wbr 4453    X. cxp 5003   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   0cc0 9504   1c1 9505   2c2 10597   #chash 12385   SPaths cspath 24315   SPathOn cspthon 24319   2SPathOnOt c2pthonot 24671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-wlk 24322  df-trail 24323  df-pth 24324  df-spth 24325  df-wlkon 24328  df-spthon 24331  df-2spthonot 24674
This theorem is referenced by:  usg2spthonot1  24704
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