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Theorem rgraprop 25133
Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rgraprop  |-  ( <. V ,  E >. RegGrph  K  ->  ( K  e.  NN0  /\ 
A. p  e.  V  ( ( V VDeg  E
) `  p )  =  K ) )
Distinct variable groups:    E, p    K, p    V, p

Proof of Theorem rgraprop
Dummy variables  e 
k  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rgra 25129 . . . 4  |- RegGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) }
21breqi 4445 . . 3  |-  ( <. V ,  E >. RegGrph  K  <->  <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) } K )
3 oprabv 6318 . . 3  |-  ( <. V ,  E >. {
<. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) } K  ->  ( V  e.  _V  /\  E  e. 
_V  /\  K  e.  _V ) )
42, 3sylbi 195 . 2  |-  ( <. V ,  E >. RegGrph  K  ->  ( V  e.  _V  /\  E  e.  _V  /\  K  e.  _V )
)
5 isrgra 25131 . . 3  |-  ( ( 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 ) ) )
65biimpd 207 . 2  |-  ( ( 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 ) ) )
74, 6mpcom 36 1  |-  ( <. V ,  E >. RegGrph  K  ->  ( 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 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106   <.cop 4022   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   {coprab 6271   NN0cn0 10791   VDeg cvdg 25098   RegGrph crgra 25127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-rgra 25129
This theorem is referenced by:  0eusgraiff0rgra  25144
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