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Theorem rusgraprop 24591
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 24587 . . . 4  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
21breqi 4446 . . 3  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } K )
3 oprabv 6320 . . 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 24589 . . 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 36 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 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106   <.cop 4026   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   {coprab 6276   NN0cn0 10784   USGrph cusg 23993   VDeg cvdg 24555   RegGrph crgra 24584   RegUSGrph crusgra 24585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-iota 5542  df-fv 5587  df-ov 6278  df-oprab 6279  df-rgra 24586  df-rusgra 24587
This theorem is referenced by:  rusisusgra  24593  cusgraiffrusgra  24602  rusgraprop2  24604  rusgranumwlks  24618  rusgranumwlk  24619  frrusgraord  24734  numclwwlk3  24772  numclwwlk5lem  24774  numclwwlk5  24775  numclwwlk7  24777  frgrareggt1  24779  frgrareg  24780  frgraregord013  24781
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