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Theorem 2spthsot 25595
Description: The set of simple paths of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
Assertion
Ref Expression
2spthsot  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2SPathOnOt  E )  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) } )
Distinct variable groups:    E, a,
b, t    V, a,
b, t
Allowed substitution hints:    X( t, a, b)    Y( t, a, b)

Proof of Theorem 2spthsot
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3089 . . 3  |-  ( V  e.  X  ->  V  e.  _V )
21adantr 466 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
3 elex 3089 . . 3  |-  ( E  e.  Y  ->  E  e.  _V )
43adantl 467 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
5 3xpexg 6609 . . . 4  |-  ( V  e.  X  ->  (
( V  X.  V
)  X.  V )  e.  _V )
65adantr 466 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  X.  V )  X.  V
)  e.  _V )
7 rabexg 4574 . . 3  |-  ( ( ( V  X.  V
)  X.  V )  e.  _V  ->  { t  e.  ( ( V  X.  V )  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) }  e.  _V )
86, 7syl 17 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) }  e.  _V )
9 id 22 . . . . . . 7  |-  ( v  =  V  ->  v  =  V )
109, 9xpeq12d 4878 . . . . . 6  |-  ( v  =  V  ->  (
v  X.  v )  =  ( V  X.  V ) )
1110, 9xpeq12d 4878 . . . . 5  |-  ( v  =  V  ->  (
( v  X.  v
)  X.  v )  =  ( ( V  X.  V )  X.  V ) )
1211adantr 466 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v  X.  v )  X.  v
)  =  ( ( V  X.  V )  X.  V ) )
13 simpl 458 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
14 oveq12 6315 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v 2SPathOnOt  e )  =  ( V 2SPathOnOt  E ) )
1514oveqd 6323 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v 2SPathOnOt  e ) b )  =  ( a ( V 2SPathOnOt  E ) b ) )
1615eleq2d 2492 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( t  e.  ( a ( v 2SPathOnOt  e
) b )  <->  t  e.  ( a ( V 2SPathOnOt  E ) b ) ) )
1713, 16rexeqbidv 3037 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e
) b )  <->  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) ) )
1813, 17rexeqbidv 3037 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e
) b )  <->  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) ) )
1912, 18rabeqbidv 3075 . . 3  |-  ( ( v  =  V  /\  e  =  E )  ->  { t  e.  ( ( v  X.  v
)  X.  v )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e
) b ) }  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) } )
20 df-2spthsot 25588 . . 3  |- 2SPathOnOt  =  ( v  e.  _V , 
e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e )
b ) } )
2119, 20ovmpt2ga 6441 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  {
t  e.  ( ( V  X.  V )  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  ( a
( V 2SPathOnOt  E ) b ) }  e.  _V )  ->  ( V 2SPathOnOt  E )  =  { t  e.  ( ( V  X.  V )  X.  V
)  |  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) } )
222, 4, 8, 21syl3anc 1264 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V 2SPathOnOt  E )  =  { t  e.  ( ( V  X.  V
)  X.  V )  |  E. a  e.  V  E. b  e.  V  t  e.  ( a ( V 2SPathOnOt  E ) b ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2772   {crab 2775   _Vcvv 3080    X. cxp 4851  (class class class)co 6306   2SPathOnOt c2spthot 25583   2SPathOnOt c2pthonot 25584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-2spthsot 25588
This theorem is referenced by:  el2spthsoton  25606  2spot0  25791
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