Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rusgrargra Structured version   Unicode version

Theorem rusgrargra 30688
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 30683 . . . 4  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
21breqi 4399 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } K )
3 oprabv 30298 . . 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 30685 . . 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 30684 . . . . . 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 1006 . . . 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 31 . . 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 36 1  |-  ( <. V ,  E >. RegUSGrph  K  -> 
<. V ,  E >. RegGrph  K
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795   _Vcvv 3071   <.cop 3984   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   {coprab 6194   NN0cn0 10683   USGrph cusg 23409   VDeg cvdg 23708   RegGrph crgra 30680   RegUSGrph crusgra 30681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-iota 5482  df-fv 5527  df-ov 6196  df-oprab 6197  df-rgra 30682  df-rusgra 30683
This theorem is referenced by:  rusgra0edg  30714
  Copyright terms: Public domain W3C validator