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Theorem rgraprop 24604
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 24600 . . . 4  |- RegGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) }
21breqi 4453 . . 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 6327 . . 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 24602 . . 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 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   VDeg cvdg 24569   RegGrph crgra 24598
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
This theorem is referenced by:  0eusgraiff0rgra  24615
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