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Theorem el2spthonot0 30395
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 30394 . 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 5696 . . . . . . . . . . . 12  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  t )  =  ( 1st `  <. A ,  b ,  C >. ) )
32fveq2d 5700 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  <. A ,  b ,  C >. )
) )
43eqeq1d 2451 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 1st `  ( 1st `  t ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  b ,  C >. ) )  =  A ) )
52fveq2d 5700 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
) )
65eqeq1d 2451 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
7 fveq2 5696 . . . . . . . . . . 11  |-  ( t  =  <. A ,  b ,  C >.  ->  ( 2nd `  t )  =  ( 2nd `  <. A ,  b ,  C >. ) )
87eqeq1d 2451 . . . . . . . . . 10  |-  ( t  =  <. A ,  b ,  C >.  ->  (
( 2nd `  t
)  =  C  <->  ( 2nd ` 
<. A ,  b ,  C >. )  =  C ) )
94, 6, 83anbi123d 1289 . . . . . . . . 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 1295 . . . . . . . 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 1682 . . . . . . 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 3122 . . . . . 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 30390 . . . . . . 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 2510 . . . . 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 994 . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . 15  |-  ( A  =  ( p ` 
0 )  ->  (
p `  0 )  =  A )
20193ad2ant1 1009 . . . . . . . . . . . . . 14  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
0 )  =  A )
21203ad2ant3 1011 . . . . . . . . . . . . 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 5696 . . . . . . . . . . . . . . 15  |-  ( (
# `  f )  =  2  ->  (
p `  ( # `  f
) )  =  ( p `  2 ) )
24 id 22 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  2 )  =  C  ->  (
p `  2 )  =  C )
2524eqcoms 2446 . . . . . . . . . . . . . . . 16  |-  ( C  =  ( p ` 
2 )  ->  (
p `  2 )  =  C )
26253ad2ant3 1011 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( p ` 
2 )  =  C )
2723, 26sylan9eq 2495 . . . . . . . . . . . . . 14  |-  ( ( ( # `  f
)  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) ) )  ->  (
p `  ( # `  f
) )  =  C )
28273adant1 1006 . . . . . . . . . . . . 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 1168 . . . . . . . . . . 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 995 . . . . . . . . . . 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 2444 . . . . . . . . . . . . . 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 30134 . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . 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 2452 . . . . . . . . . . . . . . . 16  |-  ( b  =  ( p ` 
1 )  ->  (
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b  <->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  ( p `  1 ) ) )
43423ad2ant2 1010 . . . . . . . . . . . . . . 15  |-  ( ( A  =  ( p `
 0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2 ) )  ->  ( ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b  <-> 
( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  ( p `
 1 ) ) )
44433ad2ant3 1011 . . . . . . . . . . . . . 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 1294 . . . . . . . . . . . 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 1168 . . . . . . . . . 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 1072 . . . . . . . . . . 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 995 . . . . . . . . . . 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 2451 . . . . . . . . . . . . . . . 16  |-  ( (
# `  f )  =  2  ->  (
( p `  ( # `
 f ) )  =  C  <->  ( p `  2 )  =  C ) )
52513anbi3d 1295 . . . . . . . . . . . . . . 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 5706 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 0 )  e. 
_V
54 eleq1 2503 . . . . . . . . . . . . . . . . . . . . . . . . 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 2980 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  b  e. 
_V
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  b  e.  _V )
59 fvex 5706 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( p `
 2 )  e. 
_V
60 eleq1 2503 . . . . . . . . . . . . . . . . . . . . . . . . 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 6597 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A  e.  _V  /\  b  e.  _V  /\  C  e.  _V )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. ) )  =  b )
6456, 58, 62, 63syl3anc 1218 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( p `  0
)  =  A  /\  ( p `  2
)  =  C )  ->  ( 2nd `  ( 1st `  <. A ,  b ,  C >. )
)  =  b )
6564eqeq1d 2451 . . . . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . . . 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 1006 . . . . . . . . . . . . . . . 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 1181 . . . . . . . . . . . 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 1168 . . . . . . . . . 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 828 . . . . . . . . 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 2980 . . . . . . . . . . . . 13  |-  f  e. 
_V
90 vex 2980 . . . . . . . . . . . . 13  |-  p  e. 
_V
9189, 90pm3.2i 455 . . . . . . . . . . . 12  |-  ( f  e.  _V  /\  p  e.  _V )
92 isspthonpth 23488 . . . . . . . . . . . 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 1302 . . . . . . . . . . 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 1293 . . . . . . . 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 1682 . . . . . 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 2444 . . . . . . . 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 30133 . . . . . . . 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 1220 . . . . . . 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 2737 . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2721   {crab 2724   _Vcvv 2977   <.cotp 3890   class class class wbr 4297    X. cxp 4843   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   0cc0 9287   1c1 9288   2c2 10376   #chash 12108   SPaths cspath 23413   SPathOn cspthon 23417   2SPathOnOt c2pthonot 30381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-ot 3891  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-word 12234  df-wlk 23420  df-trail 23421  df-pth 23422  df-spth 23423  df-wlkon 23426  df-spthon 23429  df-2spthonot 30384
This theorem is referenced by:  usg2spthonot1  30414
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