MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2spthsot Structured version   Unicode version

Theorem 2spthsot 24846
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 3104 . . 3  |-  ( V  e.  X  ->  V  e.  _V )
21adantr 465 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
3 elex 3104 . . 3  |-  ( E  e.  Y  ->  E  e.  _V )
43adantl 466 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
5 3xpexg 6588 . . . 4  |-  ( V  e.  X  ->  (
( V  X.  V
)  X.  V )  e.  _V )
65adantr 465 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( ( V  X.  V )  X.  V
)  e.  _V )
7 rabexg 4587 . . 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 16 . 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 5014 . . . . . 6  |-  ( v  =  V  ->  (
v  X.  v )  =  ( V  X.  V ) )
1110, 9xpeq12d 5014 . . . . 5  |-  ( v  =  V  ->  (
( v  X.  v
)  X.  v )  =  ( ( V  X.  V )  X.  V ) )
1211adantr 465 . . . 4  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v  X.  v )  X.  v
)  =  ( ( V  X.  V )  X.  V ) )
13 simpl 457 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  v  =  V )
14 oveq12 6290 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v 2SPathOnOt  e )  =  ( V 2SPathOnOt  E ) )
1514oveqd 6298 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( a ( v 2SPathOnOt  e ) b )  =  ( a ( V 2SPathOnOt  E ) b ) )
1615eleq2d 2513 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( t  e.  ( a ( v 2SPathOnOt  e
) b )  <->  t  e.  ( a ( V 2SPathOnOt  E ) b ) ) )
1713, 16rexeqbidv 3055 . . . . 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 3055 . . . 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 3090 . . 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 24839 . . 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 6417 . 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 1229 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 369    = wceq 1383    e. wcel 1804   E.wrex 2794   {crab 2797   _Vcvv 3095    X. cxp 4987  (class class class)co 6281   2SPathOnOt c2spthot 24834   2SPathOnOt c2pthonot 24835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-2spthsot 24839
This theorem is referenced by:  el2spthsoton  24857  2spot0  25042
  Copyright terms: Public domain W3C validator