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Theorem 2wlkonot3v 25603
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 3737 . . 3  |-  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( A ( V 2WalksOnOt  E ) C )  =/=  (/) )
2 df-ov 6293 . . . . 5  |-  ( A ( V 2WalksOnOt  E ) C )  =  ( ( V 2WalksOnOt  E ) `  <. A ,  C >. )
3 ndmfv 5889 . . . . 5  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( ( V 2WalksOnOt  E ) `  <. A ,  C >. )  =  (/) )
42, 3syl5eq 2497 . . . 4  |-  ( -. 
<. A ,  C >.  e. 
dom  ( V 2WalksOnOt  E )  ->  ( A ( V 2WalksOnOt  E ) C )  =  (/) )
54necon1ai 2651 . . 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 22 . . . . . . . . . . . . 13  |-  ( v  =  V  ->  v  =  V )
87, 7xpeq12d 4859 . . . . . . . . . . . 12  |-  ( v  =  V  ->  (
v  X.  v )  =  ( V  X.  V ) )
98, 7xpeq12d 4859 . . . . . . . . . . 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 6299 . . . . . . . . . . . . . 14  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v WalkOn  e )  =  ( V WalkOn  E
) )
1211oveqd 6307 . . . . . . . . . . . . 13  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v WalkOn 
e ) b )  =  ( a ( V WalkOn  E ) b ) )
1312breqd 4413 . . . . . . . . . . . 12  |-  ( ( v  =  V  /\  e  =  E )  ->  ( f ( a ( v WalkOn  e ) b ) p  <->  f (
a ( V WalkOn  E
) b ) p ) )
14133anbi1d 1343 . . . . . . . . . . 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 1770 . . . . . . . . . 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 3040 . . . . . . . . 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 6353 . . . . . . . 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 25586 . . . . . . . 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 6426 . . . . . . 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 5037 . . . . . 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 2514 . . . . 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 6378 . . . . . . . . 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 3428 . . . . . . . 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 4864 . . . . . . . . . . . 12  |-  ( <. A ,  C >.  e.  ( V  X.  V
)  <->  ( A  e.  V  /\  C  e.  V ) )
25 2wlkonot 25593 . . . . . . . . . . . . . . 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 2514 . . . . . . . . . . . . . 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 3193 . . . . . . . . . . . . . . 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 1188 . . . . . . . . . . . . . . . 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 33 . . . . . . . . . . . . . 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 32 . . . . . . . . . 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 1028 . . . . . . . . 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 32 . . . . . . . 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 6295 . . . . . . . 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 5036 . . . . . . 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 2547 . . . . . 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 32 . . . . 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 1002 . . . . 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 986 . . . . . 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 6510 . . . . . . . . . . . 12  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( V 2WalksOnOt  E )  =  (/) )
5251dmeqd 5037 . . . . . . . . . . 11  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  dom  ( V 2WalksOnOt  E )  =  dom  (/) )
5352eleq2d 2514 . . . . . . . . . 10  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( <. A ,  C >.  e.  dom  ( V 2WalksOnOt  E )  <->  <. A ,  C >.  e.  dom  (/) ) )
54 dm0 5048 . . . . . . . . . . 11  |-  dom  (/)  =  (/)
5554eleq2i 2521 . . . . . . . . . 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 3735 . . . . . . . . . 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 22 . . . . . . . . . . . . 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 6869 . . . . . . . . . . 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 981 . . . . . 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 49 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 984    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   {crab 2741   _Vcvv 3045   (/)c0 3731   <.cop 3974   class class class wbr 4402    X. cxp 4832   dom cdm 4834   ` cfv 5582  (class class class)co 6290   {coprab 6291    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792   1c1 9540   2c2 10659   #chash 12515   WalkOn cwlkon 25230   2WalksOnOt c2wlkonot 25583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-2wlkonot 25586
This theorem is referenced by:  2wlkonotv  25605  el2wlksoton  25606  frg2woteq  25788
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