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Theorem 2spthonot3v 28073
Description: If an ordered triple represents a simple path of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
2spthonot3v  |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) )

Proof of Theorem 2spthonot3v
Dummy variables  f  p  t  a  b 
c  e  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3594 . . 3  |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( A ( V 2SPathOnOt  E ) C )  =/=  (/) )
2 df-ov 6043 . . . . 5  |-  ( A ( V 2SPathOnOt  E ) C )  =  ( ( V 2SPathOnOt  E ) `  <. A ,  C >. )
3 ndmfv 5714 . . . . 5  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( ( V 2SPathOnOt  E ) `  <. A ,  C >. )  =  (/) )
42, 3syl5eq 2448 . . . 4  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( A ( V 2SPathOnOt  E ) C )  =  (/) )
54necon1ai 2609 . . 3  |-  ( ( A ( V 2SPathOnOt  E ) C )  =/=  (/)  ->  <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E ) )
6 simpl 444 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
7 id 20 . . . . . . . . . . . . 13  |-  ( v  =  V  ->  v  =  V )
87, 7xpeq12d 4862 . . . . . . . . . . . 12  |-  ( v  =  V  ->  (
v  X.  v )  =  ( V  X.  V ) )
98, 7xpeq12d 4862 . . . . . . . . . . 11  |-  ( v  =  V  ->  (
( v  X.  v
)  X.  v )  =  ( ( V  X.  V )  X.  V ) )
109adantr 452 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v  X.  v )  X.  v
)  =  ( ( V  X.  V )  X.  V ) )
11 oveq12 6049 . . . . . . . . . . . . . 14  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v SPathOn  e )  =  ( V SPathOn  E
) )
1211oveqd 6057 . . . . . . . . . . . . 13  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v SPathOn 
e ) b )  =  ( a ( V SPathOn  E ) b ) )
1312breqd 4183 . . . . . . . . . . . 12  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v SPathOn  e ) b ) p  <->  f (
a ( V SPathOn  E
) b ) p ) )
14133anbi1d 1258 . . . . . . . . . . 11  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( a ( v SPathOn  e
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) )  <->  ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) ) )
15142exbidv 1635 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E. f E. p ( f ( a ( v SPathOn  e
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) )  <->  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) ) )
1610, 15rabeqbidv 2911 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  { t  e.  ( ( v  X.  v
)  X.  v )  |  E. f E. p ( f ( a ( v SPathOn  e
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) }  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )
176, 6, 16mpt2eq123dv 6095 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a  e.  v ,  b  e.  v 
|->  { t  e.  ( ( v  X.  v
)  X.  v )  |  E. f E. p ( f ( a ( v SPathOn  e
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
18 df-2spthonot 28057 . . . . . . . 8  |- 2SPathOnOt  =  ( v  e.  _V , 
e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { t  e.  ( ( v  X.  v )  X.  v )  |  E. f E. p
( f ( a ( v SPathOn  e ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
1917, 18ovmpt2ga 6162 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( V 2SPathOnOt  E )  =  ( a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
2019dmeqd 5031 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  dom  ( V 2SPathOnOt  E )  =  dom  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
2120eleq2d 2471 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  <->  <. A ,  C >.  e. 
dom  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) ) )
22 dmoprabss 6114 . . . . . . . . 9  |-  dom  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  V  /\  b  e.  V )  /\  c  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) } 
C_  ( V  X.  V )
2322sseli 3304 . . . . . . . 8  |-  ( <. A ,  C >.  e. 
dom  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  V  /\  b  e.  V )  /\  c  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) }  ->  <. A ,  C >.  e.  ( V  X.  V ) )
24 opelxp 4867 . . . . . . . . . . . 12  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  <->  ( A  e.  V  /\  C  e.  V ) )
25 2spthonot 28063 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( 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 ) ) } )
2625eleq2d 2471 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  <->  T  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 ) ) } ) )
27 elrabi 3050 . . . . . . . . . . . . . . 15  |-  ( T  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 ) ) }  ->  T  e.  ( ( V  X.  V )  X.  V
) )
28 simpl 444 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
2928adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  T  e.  ( ( V  X.  V )  X.  V
) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
30 simpr 448 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( A  e.  V  /\  C  e.  V )
)
3130adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  T  e.  ( ( V  X.  V )  X.  V
) )  ->  ( A  e.  V  /\  C  e.  V )
)
32 simpr 448 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  T  e.  ( ( V  X.  V )  X.  V
) )  ->  T  e.  ( ( V  X.  V )  X.  V
) )
3329, 31, 323jca 1134 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )
)  /\  T  e.  ( ( V  X.  V )  X.  V
) )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) )
3433ex 424 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( ( V  X.  V )  X.  V )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) )
3527, 34syl5 30 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  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 ) ) }  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) )
3626, 35sylbid 207 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
3736expcom 425 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
3824, 37sylbi 188 . . . . . . . . . . 11  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
3938com12 29 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  ( V  X.  V )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
40393adant3 977 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e.  ( V  X.  V
)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
4140com12 29 . . . . . . . 8  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V  /\  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  e.  _V )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) ) )
4223, 41syl 16 . . . . . . 7  |-  ( <. A ,  C >.  e. 
dom  { <. <. a ,  b
>. ,  c >.  |  ( ( a  e.  V  /\  b  e.  V )  /\  c  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) }  ->  ( ( V  e.  _V  /\  E  e.  _V  /\  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  e.  _V )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) ) )
43 df-mpt2 6045 . . . . . . . 8  |-  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  =  { <. <.
a ,  b >. ,  c >.  |  ( ( a  e.  V  /\  b  e.  V
)  /\  c  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) }
4443dmeqi 5030 . . . . . . 7  |-  dom  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  =  dom  { <. <. a ,  b >. ,  c
>.  |  ( (
a  e.  V  /\  b  e.  V )  /\  c  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) }
4542, 44eleq2s 2496 . . . . . 6  |-  ( <. A ,  C >.  e. 
dom  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  -> 
( ( V  e. 
_V  /\  E  e.  _V  /\  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
4645com12 29 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  -> 
( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) ) )
4721, 46sylbid 207 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
48 3ianor 951 . . . . 5  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  <->  ( -.  V  e.  _V  \/  -.  E  e.  _V  \/  -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V ) )
49 df-3or 937 . . . . . 6  |-  ( ( -.  V  e.  _V  \/  -.  E  e.  _V  \/  -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  <->  ( ( -.  V  e.  _V  \/  -.  E  e.  _V )  \/  -.  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V SPathOn  E )
b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V ) )
50 ianor 475 . . . . . . . . 9  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  <->  ( -.  V  e.  _V  \/  -.  E  e.  _V ) )
5118mpt2ndm0 6432 . . . . . . . . . . . . 13  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( V 2SPathOnOt  E )  =  (/) )
5251dmeqd 5031 . . . . . . . . . . . 12  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  dom  ( V 2SPathOnOt  E )  =  dom  (/) )
5352eleq2d 2471 . . . . . . . . . . 11  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  <->  <. A ,  C >.  e.  dom  (/) ) )
54 dm0 5042 . . . . . . . . . . . 12  |-  dom  (/)  =  (/)
5554eleq2i 2468 . . . . . . . . . . 11  |-  ( <. A ,  C >.  e. 
dom  (/)  <->  <. A ,  C >.  e.  (/) )
5653, 55syl6bb 253 . . . . . . . . . 10  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  <->  <. A ,  C >.  e.  (/) ) )
57 noel 3592 . . . . . . . . . . 11  |-  -.  <. A ,  C >.  e.  (/)
5857pm2.21i 125 . . . . . . . . . 10  |-  ( <. A ,  C >.  e.  (/)  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
5956, 58syl6bi 220 . . . . . . . . 9  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
6050, 59sylbir 205 . . . . . . . 8  |-  ( ( -.  V  e.  _V  \/  -.  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
61 anor 476 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  <->  -.  ( -.  V  e. 
_V  \/  -.  E  e.  _V ) )
62 id 20 . . . . . . . . . . . . . 14  |-  ( V  e.  _V  ->  V  e.  _V )
6362ancri 536 . . . . . . . . . . . . 13  |-  ( V  e.  _V  ->  ( V  e.  _V  /\  V  e.  _V ) )
6463adantr 452 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  V  e.  _V )
)
65 mpt2exga 6383 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  V  e.  _V )  ->  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )
6664, 65syl 16 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )
6766pm2.24d 137 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  e.  _V  ->  (
<. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) ) )
6861, 67sylbir 205 . . . . . . . . 9  |-  ( -.  ( -.  V  e. 
_V  \/  -.  E  e.  _V )  ->  ( -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) ) )
6968imp 419 . . . . . . . 8  |-  ( ( -.  ( -.  V  e.  _V  \/  -.  E  e.  _V )  /\  -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7060, 69jaoi 369 . . . . . . 7  |-  ( ( ( -.  V  e. 
_V  \/  -.  E  e.  _V )  \/  ( -.  ( -.  V  e. 
_V  \/  -.  E  e.  _V )  /\  -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V ) )  -> 
( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A
( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7170jaoi2 934 . . . . . 6  |-  ( ( ( -.  V  e. 
_V  \/  -.  E  e.  _V )  \/  -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7249, 71sylbi 188 . . . . 5  |-  ( ( -.  V  e.  _V  \/  -.  E  e.  _V  \/  -.  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7348, 72sylbi 188 . . . 4  |-  ( -.  ( V  e.  _V  /\  E  e.  _V  /\  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. f E. p ( f ( a ( V SPathOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7447, 73pm2.61i 158 . . 3  |-  ( <. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
751, 5, 743syl 19 . 2  |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
7675pm2.43i 45 1  |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670   _Vcvv 2916   (/)c0 3588   <.cop 3777   class class class wbr 4172    X. cxp 4835   dom cdm 4837   ` cfv 5413  (class class class)co 6040   {coprab 6041    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   1c1 8947   2c2 10005   #chash 11573   SPathOn cspthon 21466   2SPathOnOt c2pthonot 28054
This theorem is referenced by:  el2spthsoton  28076  2spontn0vne  28084  2spot0  28167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-2spthonot 28057
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