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

Theorem rusgraprop 25346
Description: The properties of a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusgraprop  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  (
( V VDeg  E ) `  p )  =  K ) )
Distinct variable groups:    E, p    K, p    V, p

Proof of Theorem rusgraprop
Dummy variables  e 
k  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgra 25342 . . . 4  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
21breqi 4401 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } K )
3 oprabv 6326 . . 3  |-  ( <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } K  -> 
( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )
)
42, 3sylbi 195 . 2  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )
)
5 isrusgra 25344 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )  ->  ( <. V ,  E >. RegUSGrph  K  <->  ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K ) ) )
65biimpd 207 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )  ->  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  (
( V VDeg  E ) `  p )  =  K ) ) )
74, 6mpcom 34 1  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  (
( V VDeg  E ) `  p )  =  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   _Vcvv 3059   <.cop 3978   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   {coprab 6279   NN0cn0 10836   USGrph cusg 24747   VDeg cvdg 25310   RegGrph crgra 25339   RegUSGrph crusgra 25340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-iota 5533  df-fv 5577  df-ov 6281  df-oprab 6282  df-rgra 25341  df-rusgra 25342
This theorem is referenced by:  rusisusgra  25348  cusgraiffrusgra  25357  rusgraprop2  25359  rusgranumwlks  25373  rusgranumwlk  25374  frrusgraord  25488  numclwwlk3  25526  numclwwlk5lem  25528  numclwwlk5  25529  numclwwlk7  25531  frgrareggt1  25533  frgrareg  25534  frgraregord013  25535
  Copyright terms: Public domain W3C validator