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Theorem 2wlkonot3v 25595
Description: If an ordered triple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
Assertion
Ref Expression
2wlkonot3v  |-  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) )

Proof of Theorem 2wlkonot3v
Dummy variables  f  p  t  a  b 
c  e  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3768 . . 3  |-  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( A ( V 2WalksOnOt  E ) C )  =/=  (/) )
2 df-ov 6306 . . . . 5  |-  ( A ( V 2WalksOnOt  E ) C )  =  ( ( V 2WalksOnOt  E ) `  <. A ,  C >. )
3 ndmfv 5903 . . . . 5  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( ( V 2WalksOnOt  E ) `  <. A ,  C >. )  =  (/) )
42, 3syl5eq 2476 . . . 4  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( A ( V 2WalksOnOt  E ) C )  =  (/) )
54necon1ai 2656 . . 3  |-  ( ( A ( V 2WalksOnOt  E ) C )  =/=  (/)  ->  <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E ) )
6 simpl 459 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
7 id 23 . . . . . . . . . . . . 13  |-  ( v  =  V  ->  v  =  V )
87, 7xpeq12d 4876 . . . . . . . . . . . 12  |-  ( v  =  V  ->  (
v  X.  v )  =  ( V  X.  V ) )
98, 7xpeq12d 4876 . . . . . . . . . . 11  |-  ( v  =  V  ->  (
( v  X.  v
)  X.  v )  =  ( ( V  X.  V )  X.  V ) )
109adantr 467 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v  X.  v )  X.  v
)  =  ( ( V  X.  V )  X.  V ) )
11 oveq12 6312 . . . . . . . . . . . . . 14  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1211oveqd 6320 . . . . . . . . . . . . 13  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1312breqd 4432 . . . . . . . . . . . 12  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
14133anbi1d 1340 . . . . . . . . . . 11  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( f ( a ( v WalkOn  e
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) )  <->  ( f ( a ( V WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) ) )
15142exbidv 1761 . . . . . . . . . 10  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E. f E. p ( f ( a ( v WalkOn  e
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) )  <->  E. f E. p
( f ( a ( V WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) ) )
1610, 15rabeqbidv 3077 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  { t  e.  ( ( v  X.  v
)  X.  v )  |  E. f E. p ( f ( a ( v WalkOn  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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )
176, 6, 16mpt2eq123dv 6365 . . . . . . . 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 WalkOn  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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
18 df-2wlkonot 25578 . . . . . . . 8  |- 2WalksOnOt  =  ( 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 WalkOn  e ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
1917, 18ovmpt2ga 6438 . . . . . . 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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( V 2WalksOnOt  E )  =  ( a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
2019dmeqd 5054 . . . . . 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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  dom  ( V 2WalksOnOt  E )  =  dom  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( a ( V WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) )
2120eleq2d 2493 . . . . 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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) ) )
22 dmoprabss 6390 . . . . . . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) } 
C_  ( V  X.  V )
2322sseli 3461 . . . . . . . 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 WalkOn  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 4881 . . . . . . . . . . . 12  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  <->  ( A  e.  V  /\  C  e.  V ) )
25 2wlkonot 25585 . . . . . . . . . . . . . . 15  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( A ( V 2WalksOnOt  E ) C )  =  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( A ( V WalkOn  E ) C ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  C ) ) } )
2625eleq2d 2493 . . . . . . . . . . . . . 14  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  <->  T  e.  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( A ( V WalkOn  E ) C ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  C ) ) } ) )
27 elrabi 3227 . . . . . . . . . . . . . . 15  |-  ( T  e.  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. f E. p ( f ( A ( V WalkOn  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 459 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( V  e.  _V  /\  E  e.  _V ) )
2928adantr 467 . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( A  e.  V  /\  C  e.  V )
)
3130adantr 467 . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . 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 34 . . . . . . . . . . . . . 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 WalkOn  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 219 . . . . . . . . . . . . 13  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
) )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
3736expcom 437 . . . . . . . . . . . 12  |-  ( ( A  e.  V  /\  C  e.  V )  ->  ( ( V  e. 
_V  /\  E  e.  _V )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
3824, 37sylbi 199 . . . . . . . . . . 11  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
3938com12 33 . . . . . . . . . 10  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  ( V  X.  V )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
40393adant3 1026 . . . . . . . . 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 WalkOn  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 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
4140com12 33 . . . . . . . 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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  e.  _V )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) ) )
4223, 41syl 17 . . . . . . 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 WalkOn  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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  e.  _V )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) ) )
43 df-mpt2 6308 . . . . . . . 8  |-  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p ( f ( a ( V WalkOn  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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) }
4443dmeqi 5053 . . . . . . 7  |-  dom  (
a  e.  V , 
b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. f E. p
( f ( a ( V WalkOn  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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } ) }
4542, 44eleq2s 2531 . . . . . 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 WalkOn  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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
4645com12 33 . . . . 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 WalkOn  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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  -> 
( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V
)  /\  T  e.  ( ( V  X.  V )  X.  V
) ) ) ) )
4721, 46sylbid 219 . . . 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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
48 3ianor 1000 . . . . 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 WalkOn  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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V ) )
49 df-3or 984 . . . . . 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 WalkOn  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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  (
( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V ) )
50 ianor 491 . . . . . . . 8  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  <->  ( -.  V  e.  _V  \/  -.  E  e.  _V ) )
5118mpt2ndm0 6522 . . . . . . . . . . . 12  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( V 2WalksOnOt  E )  =  (/) )
5251dmeqd 5054 . . . . . . . . . . 11  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  dom  ( V 2WalksOnOt  E )  =  dom  (/) )
5352eleq2d 2493 . . . . . . . . . 10  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E )  <->  <. A ,  C >.  e.  dom  (/) ) )
54 dm0 5065 . . . . . . . . . . 11  |-  dom  (/)  =  (/)
5554eleq2i 2501 . . . . . . . . . 10  |-  ( <. A ,  C >.  e. 
dom  (/)  <->  <. A ,  C >.  e.  (/) )
5653, 55syl6bb 265 . . . . . . . . 9  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E )  <->  <. A ,  C >.  e.  (/) ) )
57 noel 3766 . . . . . . . . . 10  |-  -.  <. A ,  C >.  e.  (/)
5857pm2.21i 135 . . . . . . . . 9  |-  ( <. A ,  C >.  e.  (/)  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
5956, 58syl6bi 232 . . . . . . . 8  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
6050, 59sylbir 217 . . . . . . 7  |-  ( ( -.  V  e.  _V  \/  -.  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
61 anor 492 . . . . . . . . 9  |-  ( ( V  e.  _V  /\  E  e.  _V )  <->  -.  ( -.  V  e. 
_V  \/  -.  E  e.  _V ) )
62 id 23 . . . . . . . . . . . . 13  |-  ( V  e.  _V  ->  V  e.  _V )
6362ancri 555 . . . . . . . . . . . 12  |-  ( V  e.  _V  ->  ( V  e.  _V  /\  V  e.  _V ) )
6463adantr 467 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V  e.  _V  /\  V  e.  _V )
)
65 mpt2exga 6881 . . . . . . . . . . 11  |-  ( ( 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )
6664, 65syl 17 . . . . . . . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )
6766pm2.24d 138 . . . . . . . . 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 WalkOn  E ) b ) p  /\  ( # `  f )  =  2  /\  ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b ) ) } )  e.  _V  ->  (
<. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) ) )
6861, 67sylbir 217 . . . . . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V  ->  ( <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A
( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) ) )
6968imp 431 . . . . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7060, 69jaoi3 979 . . . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7149, 70sylbi 199 . . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7248, 71sylbi 199 . . . 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 WalkOn  E
) b ) p  /\  ( # `  f
)  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
) )  =  ( p `  1 )  /\  ( 2nd `  t
)  =  b ) ) } )  e. 
_V )  ->  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) ) )
7347, 72pm2.61i 168 . . 3  |-  ( <. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
741, 5, 733syl 18 . 2  |-  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  (
( V  X.  V
)  X.  V ) ) ) )
7574pm2.43i 50 1  |-  ( T  e.  ( A ( V 2WalksOnOt  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 370    /\ wa 371    \/ w3o 982    /\ w3a 983    = wceq 1438   E.wex 1660    e. wcel 1869    =/= wne 2619   {crab 2780   _Vcvv 3082   (/)c0 3762   <.cop 4003   class class class wbr 4421    X. cxp 4849   dom cdm 4851   ` cfv 5599  (class class class)co 6303   {coprab 6304    |-> cmpt2 6305   1stc1st 6803   2ndc2nd 6804   1c1 9542   2c2 10661   #chash 12516   WalkOn cwlkon 25222   2WalksOnOt c2wlkonot 25575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-2wlkonot 25578
This theorem is referenced by:  2wlkonotv  25597  el2wlksoton  25598  frg2woteq  25780
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