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

Theorem rusgrargra 25228
Description: A k-regular undirected simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusgrargra  |-  ( <. V ,  E >. RegUSGrph  K  -> 
<. V ,  E >. RegGrph  K
)

Proof of Theorem rusgrargra
Dummy variables  e 
k  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgra 25223 . . . 4  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
21breqi 4400 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } K )
3 oprabv 6282 . . 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 25225 . . 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 ) ) )
6 isrgra 25224 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )  ->  ( <. V ,  E >. RegGrph  K  <->  ( K  e.  NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `
 p )  =  K ) ) )
76biimprcd 225 . . . . 5  |-  ( ( K  e.  NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `
 p )  =  K )  ->  (
( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )  -> 
<. V ,  E >. RegGrph  K
) )
873adant1 1015 . . . 4  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K )  ->  ( ( V  e.  _V  /\  E  e.  _V  /\  K  e. 
_V )  ->  <. V ,  E >. RegGrph  K ) )
98com12 29 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )  ->  (
( V USGrph  E  /\  K  e.  NN0  /\  A. p  e.  V  (
( V VDeg  E ) `  p )  =  K )  ->  <. V ,  E >. RegGrph  K ) )
105, 9sylbid 215 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )  ->  ( <. V ,  E >. RegUSGrph  K  -> 
<. V ,  E >. RegGrph  K
) )
114, 10mpcom 34 1  |-  ( <. V ,  E >. RegUSGrph  K  -> 
<. V ,  E >. RegGrph  K
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   _Vcvv 3058   <.cop 3977   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   {coprab 6235   NN0cn0 10756   USGrph cusg 24628   VDeg cvdg 25191   RegGrph crgra 25220   RegUSGrph crusgra 25221
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 4516  ax-nul 4524  ax-pr 4629
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 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-iota 5489  df-fv 5533  df-ov 6237  df-oprab 6238  df-rgra 25222  df-rusgra 25223
This theorem is referenced by:  rusgra0edg  25253
  Copyright terms: Public domain W3C validator