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Theorem el2spthonot0 25300
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 25299 . 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 5851 . . . . . . . . . . . 12  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  t )  =  ( 1st `  <. A ,  b ,  C >. ) )
32fveq2d 5855 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  <. A ,  b ,  C >. )
) )
43eqeq1d 2406 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A ) )
52fveq2d 5855 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
) )
65eqeq1d 2406 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
7 fveq2 5851 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  t )  =  ( 2nd `  <. A ,  b ,  C >. ) )
87eqeq1d 2406 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd ` 
<. A ,  b ,  C >. )  =  C ) )
94, 6, 83anbi123d 1303 . . . . . . . . 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 1309 . . . . . . . 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 1739 . . . . . . 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 3209 . . . . . 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 25295 . . . . . . 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 2474 . . . . 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 1005 . . . . . . . . . . . 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 23 . . . . . . . . . . . . . . . 16  |-  ( ( p `  0 )  =  A  ->  (
p `  0 )  =  A )
1918eqcoms 2416 . . . . . . . . . . . . . . 15  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
20193ad2ant1 1020 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
21203ad2ant3 1022 . . . . . . . . . . . . 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 5851 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
24 id 23 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  2 )  =  C  ->  (
p `  2 )  =  C )
2524eqcoms 2416 . . . . . . . . . . . . . . . 16  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
26253ad2ant3 1022 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
2723, 26sylan9eq 2465 . . . . . . . . . . . . . 14  |-  ( ( ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  (
p `  ( # `  f
) )  =  C )
28273adant1 1017 . . . . . . . . . . . . 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 1179 . . . . . . . . . . 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 1006 . . . . . . . . . . 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 2405 . . . . . . . . . . . . . 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 6805 . . . . . . . . . . . . . . 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 1234 . . . . . . . . . . . . 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 2419 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
43423ad2ant2 1021 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
44433ad2ant3 1022 . . . . . . . . . . . . . 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 1308 . . . . . . . . . . . 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 212 . . . . . . . . . . 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 1179 . . . . . . . . . 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 1083 . . . . . . . . . . 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 1006 . . . . . . . . . . 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 2406 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
52513anbi3d 1309 . . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 0 )  e. 
_V
54 eleq1 2476 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  0 )  =  A  ->  (
( p `  0
)  e.  _V  <->  A  e.  _V ) )
5553, 54mpbii 213 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  0 )  =  A  ->  A  e.  _V )
5655adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  A  e.  _V )
57 vex 3064 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  b  e. 
_V
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  b  e.  _V )
59 fvex 5861 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 2 )  e. 
_V
60 eleq1 2476 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( p `  2 )  =  C  ->  (
( p `  2
)  e.  _V  <->  C  e.  _V ) )
6159, 60mpbii 213 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  e.  _V )
6261adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  C  e.  _V )
63 ot2ndg 6801 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  e.  _V  /\  b  e.  _V  /\  C  e.  _V )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b )
6456, 58, 62, 63syl3anc 1232 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b )
6564eqeq1d 2406 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 )  <-> 
b  =  ( p `
 1 ) ) )
66 id 23 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( A  =  ( p ` 
0 )  ->  A  =  ( p ` 
0 ) )
6766eqcoms 2416 . . . . . . . . . . . . . . . . . . . . . . . . 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 23 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( C  =  ( p ` 
2 )  ->  C  =  ( p ` 
2 ) )
7271eqcoms 2416 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( p `  2 )  =  C  ->  C  =  ( p ` 
2 ) )
7372ad2antlr 727 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( p ` 
0 )  =  A  /\  ( p ` 
2 )  =  C )  /\  b  =  ( p `  1
) )  ->  C  =  ( p ` 
2 ) )
7469, 70, 733jca 1179 . . . . . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . 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 1179 . . . . . . . . . 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 835 . . . . . . . . 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 714 . . . . . . . 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 3064 . . . . . . . . . . . . 13  |-  f  e. 
_V
90 vex 3064 . . . . . . . . . . . . 13  |-  p  e. 
_V
9189, 90pm3.2i 455 . . . . . . . . . . . 12  |-  ( f  e.  _V  /\  p  e.  _V )
92 isspthonpth 25015 . . . . . . . . . . . 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 1316 . . . . . . . . . . 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 203 . . . . . . . . 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 1307 . . . . . . . 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 255 . . . . . . 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 1739 . . . . . 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 2405 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  <. A , 
b ,  C >.  = 
<. A ,  b ,  C >. )
10033ad2antlr 727 . . . . . . . 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 727 . . . . . . . 8  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  b  e.  V )  ->  C  e.  V )
103 otel3xp 4861 . . . . . . . 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 1234 . . . . . . 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 255 . . . . 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 288 . . . 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 704 . . 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 2917 . 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 255 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 186    /\ wa 369    /\ w3a 976    = wceq 1407   E.wex 1635    e. wcel 1844   E.wrex 2757   {crab 2760   _Vcvv 3061   <.cotp 3982   class class class wbr 4397    X. cxp 4823   ` cfv 5571  (class class class)co 6280   1stc1st 6784   2ndc2nd 6785   0cc0 9524   1c1 9525   2c2 10628   #chash 12454   SPaths cspath 24930   SPathOn cspthon 24934   2SPathOnOt c2pthonot 25286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-ot 3983  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-wlk 24937  df-trail 24938  df-pth 24939  df-spth 24940  df-wlkon 24943  df-spthon 24946  df-2spthonot 25289
This theorem is referenced by:  usg2spthonot1  25319
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