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

Theorem rusgrargra 24606
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 24601 . . . 4  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
21breqi 4453 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } K )
3 oprabv 6327 . . 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 24603 . . 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 24602 . . . . . 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 1014 . . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   {coprab 6283   NN0cn0 10791   USGrph cusg 24006   VDeg cvdg 24569   RegGrph crgra 24598   RegUSGrph crusgra 24599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-iota 5549  df-fv 5594  df-ov 6285  df-oprab 6286  df-rgra 24600  df-rusgra 24601
This theorem is referenced by:  rusgra0edg  24631
  Copyright terms: Public domain W3C validator