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Theorem 2wlkonot 24529
Description: The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
2wlkonot  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2WalksOnOt  E ) 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 ) ) } )
Distinct variable groups:    t, E, f, p    t, V, f, p    A, f, p, t    B, f, p, t
Allowed substitution hints:    X( t, f, p)    Y( t, f, p)

Proof of Theorem 2wlkonot
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2wlkonot 24527 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) ) } ) )
21adantr 465 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  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 ) ) } ) )
32oveqd 6294 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2WalksOnOt  E ) B )  =  ( A ( 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 ) ) } ) B ) )
4 simprl 755 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  A  e.  V )
5 simprr 756 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  ->  B  e.  V )
6 xpexg 6704 . . . . . . 7  |-  ( ( V  e.  X  /\  V  e.  X )  ->  ( V  X.  V
)  e.  _V )
76anidms 645 . . . . . 6  |-  ( V  e.  X  ->  ( V  X.  V )  e. 
_V )
8 xpexg 6704 . . . . . 6  |-  ( ( ( V  X.  V
)  e.  _V  /\  V  e.  X )  ->  ( ( V  X.  V )  X.  V
)  e.  _V )
97, 8mpancom 669 . . . . 5  |-  ( V  e.  X  ->  (
( V  X.  V
)  X.  V )  e.  _V )
109ad2antrr 725 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( ( V  X.  V )  X.  V
)  e.  _V )
11 rabexg 4592 . . . 4  |-  ( ( ( V  X.  V
)  X.  V )  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 )
1210, 11syl 16 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( 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 )
13 oveq12 6286 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  B )  ->  ( a ( V WalkOn  E ) b )  =  ( A ( V WalkOn  E ) B ) )
1413breqd 4453 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( f ( a ( V WalkOn  E ) b ) p  <->  f ( A ( V WalkOn  E
) B ) p ) )
15 eqeq2 2477 . . . . . . . . 9  |-  ( a  =  A  ->  (
( 1st `  ( 1st `  t ) )  =  a  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
1615adantr 465 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 1st `  ( 1st `  t ) )  =  a  <->  ( 1st `  ( 1st `  t
) )  =  A ) )
17 eqeq2 2477 . . . . . . . . 9  |-  ( b  =  B  ->  (
( 2nd `  t
)  =  b  <->  ( 2nd `  t )  =  B ) )
1817adantl 466 . . . . . . . 8  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 2nd `  t
)  =  b  <->  ( 2nd `  t )  =  B ) )
1916, 183anbi13d 1296 . . . . . . 7  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( ( 1st `  ( 1st `  t
) )  =  a  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  b )  <->  ( ( 1st `  ( 1st `  t
) )  =  A  /\  ( 2nd `  ( 1st `  t ) )  =  ( p ` 
1 )  /\  ( 2nd `  t )  =  B ) ) )
2014, 193anbi13d 1296 . . . . . 6  |-  ( ( a  =  A  /\  b  =  B )  ->  ( ( 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 ) ) ) )
21202exbidv 1687 . . . . 5  |-  ( ( a  =  A  /\  b  =  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 ) )  <->  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 ) ) ) )
2221rabbidv 3100 . . . 4  |-  ( ( a  =  A  /\  b  =  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 ) ) }  =  {
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 ) ) } )
23 eqid 2462 . . . 4  |-  ( 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 ) ) } )
2422, 23ovmpt2ga 6409 . . 3  |-  ( ( 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 ( 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 ) ) } ) 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 ) ) } )
254, 5, 12, 24syl3anc 1223 . 2  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( 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 ) ) } ) 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 ) ) } )
263, 25eqtrd 2503 1  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( A  e.  V  /\  B  e.  V ) )  -> 
( A ( V 2WalksOnOt  E ) 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 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   {crab 2813   _Vcvv 3108   class class class wbr 4442    X. cxp 4992   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   1c1 9484   2c2 10576   #chash 12362   WalkOn cwlkon 24166   2WalksOnOt c2wlkonot 24519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-2wlkonot 24522
This theorem is referenced by:  el2wlkonot  24533  2wlkonot3v  24539  2pthwlkonot  24549
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