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Theorem 2spthonot3v 30393
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 3642 . . 3  |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( A ( V 2SPathOnOt  E ) C )  =/=  (/) )
2 df-ov 6093 . . . . 5  |-  ( A ( V 2SPathOnOt  E ) C )  =  ( ( V 2SPathOnOt  E ) `  <. A ,  C >. )
3 ndmfv 5713 . . . . 5  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( ( V 2SPathOnOt  E ) `  <. A ,  C >. )  =  (/) )
42, 3syl5eq 2486 . . . 4  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2SPathOnOt  E )  ->  ( A ( V 2SPathOnOt  E ) C )  =  (/) )
54necon1ai 2652 . . 3  |-  ( ( A ( V 2SPathOnOt  E ) C )  =/=  (/)  ->  <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E ) )
6 simpl 457 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
7 id 22 . . . . . . . . . . . . 13  |-  ( v  =  V  ->  v  =  V )
87, 7xpeq12d 4864 . . . . . . . . . . . 12  |-  ( v  =  V  ->  (
v  X.  v )  =  ( V  X.  V ) )
98, 7xpeq12d 4864 . . . . . . . . . . 11  |-  ( v  =  V  ->  (
( v  X.  v
)  X.  v )  =  ( ( V  X.  V )  X.  V ) )
109adantr 465 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v  X.  v )  X.  v
)  =  ( ( V  X.  V )  X.  V ) )
11 oveq12 6099 . . . . . . . . . . . . . 14  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v SPathOn  e )  =  ( V SPathOn  E
) )
1211oveqd 6107 . . . . . . . . . . . . 13  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v SPathOn 
e ) b )  =  ( a ( V SPathOn  E ) b ) )
1312breqd 4302 . . . . . . . . . . . 12  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v SPathOn  e ) b ) p  <->  f (
a ( V SPathOn  E
) b ) p ) )
14133anbi1d 1293 . . . . . . . . . . 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 1682 . . . . . . . . . 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 2966 . . . . . . . . 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 6147 . . . . . . . 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 30377 . . . . . . . 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 6219 . . . . . . 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 5041 . . . . . 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 2509 . . . . 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 6171 . . . . . . . . 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 3351 . . . . . . . 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 4868 . . . . . . . . . . . 12  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  <->  ( A  e.  V  /\  C  e.  V ) )
25 2spthonot 30383 . . . . . . . . . . . . . . 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 2509 . . . . . . . . . . . . . 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 3113 . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
2928adantr 465 . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( A  e.  V  /\  C  e.  V )
)
3130adantr 465 . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . 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 1168 . . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . . 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 32 . . . . . . . . . . . . . 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 215 . . . . . . . . . . . . 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 435 . . . . . . . . . . . 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 195 . . . . . . . . . . 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 31 . . . . . . . . . 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 1008 . . . . . . . . 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 31 . . . . . . . 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 6095 . . . . . . . 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 5040 . . . . . . 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 2534 . . . . . 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 31 . . . . 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 215 . . . 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 982 . . . . 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 966 . . . . . 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 488 . . . . . . . . 9  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  <->  ( -.  V  e.  _V  \/  -.  E  e.  _V ) )
5118mpt2ndm0 6738 . . . . . . . . . . . . 13  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( V 2SPathOnOt  E )  =  (/) )
5251dmeqd 5041 . . . . . . . . . . . 12  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  dom  ( V 2SPathOnOt  E )  =  dom  (/) )
5352eleq2d 2509 . . . . . . . . . . 11  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  <->  <. A ,  C >.  e.  dom  (/) ) )
54 dm0 5052 . . . . . . . . . . . 12  |-  dom  (/)  =  (/)
5554eleq2i 2506 . . . . . . . . . . 11  |-  ( <. A ,  C >.  e. 
dom  (/)  <->  <. A ,  C >.  e.  (/) )
5653, 55syl6bb 261 . . . . . . . . . 10  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2SPathOnOt  E )  <->  <. A ,  C >.  e.  (/) ) )
57 noel 3640 . . . . . . . . . . 11  |-  -.  <. A ,  C >.  e.  (/)
5857pm2.21i 131 . . . . . . . . . 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 228 . . . . . . . . 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 213 . . . . . . . 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 489 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  <->  -.  ( -.  V  e. 
_V  \/  -.  E  e.  _V ) )
62 id 22 . . . . . . . . . . . . . 14  |-  ( V  e.  _V  ->  V  e.  _V )
6362ancri 552 . . . . . . . . . . . . 13  |-  ( V  e.  _V  ->  ( V  e.  _V  /\  V  e.  _V ) )
6463adantr 465 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  V  e.  _V )
)
65 mpt2exga 6648 . . . . . . . . . . . 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 143 . . . . . . . . . 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 213 . . . . . . . . 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 429 . . . . . . . 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 379 . . . . . . 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 959 . . . . . 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 195 . . . . 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 195 . . . 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 164 . . 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 20 . 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 47 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2605   {crab 2718   _Vcvv 2971   (/)c0 3636   <.cop 3882   class class class wbr 4291    X. cxp 4837   dom cdm 4839   ` cfv 5417  (class class class)co 6090   {coprab 6091    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   1c1 9282   2c2 10370   #chash 12102   SPathOn cspthon 23411   2SPathOnOt c2pthonot 30374
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-2spthonot 30377
This theorem is referenced by:  el2spthsoton  30396  2spontn0vne  30404  2spot0  30655
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