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Theorem isrgra 24749
Description: The property of being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
Assertion
Ref Expression
isrgra  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. 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
Allowed substitution hints:    X( p)    Y( p)    Z( p)

Proof of Theorem isrgra
Dummy variables  e 
k  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4454 . 2  |-  ( <. V ,  E >. RegGrph  K  <->  <. <. V ,  E >. ,  K >.  e. RegGrph  )
2 df-rgra 24747 . . . 4  |- RegGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) }
32eleq2i 2545 . . 3  |-  ( <. <. V ,  E >. ,  K >.  e. RegGrph  <->  <. <. V ,  E >. ,  K >.  e. 
{ <. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) } )
4 eleq1 2539 . . . . . 6  |-  ( k  =  K  ->  (
k  e.  NN0  <->  K  e.  NN0 ) )
543ad2ant3 1019 . . . . 5  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( k  e.  NN0  <->  K  e.  NN0 ) )
6 simp1 996 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  v  =  V )
7 oveq12 6304 . . . . . . . . 9  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v VDeg  e )  =  ( V VDeg  E
) )
87fveq1d 5874 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ( v VDeg  e
) `  p )  =  ( ( V VDeg 
E ) `  p
) )
983adant3 1016 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( v VDeg  e
) `  p )  =  ( ( V VDeg 
E ) `  p
) )
10 simp3 998 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  k  =  K )
119, 10eqeq12d 2489 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( ( v VDeg  e ) `  p
)  =  k  <->  ( ( V VDeg  E ) `  p
)  =  K ) )
126, 11raleqbidv 3077 . . . . 5  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( A. p  e.  v  ( ( v VDeg  e ) `  p
)  =  k  <->  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K ) )
135, 12anbi12d 710 . . . 4  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( k  e. 
NN0  /\  A. p  e.  v  ( (
v VDeg  e ) `  p )  =  k )  <->  ( K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K ) ) )
1413eloprabga 6384 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. <. V ,  E >. ,  K >.  e.  { <. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) }  <-> 
( K  e.  NN0  /\ 
A. p  e.  V  ( ( V VDeg  E
) `  p )  =  K ) ) )
153, 14syl5bb 257 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. <. V ,  E >. ,  K >.  e. RegGrph  <->  ( K  e.  NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K ) ) )
161, 15syl5bb 257 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. 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    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   <.cop 4039   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   {coprab 6296   NN0cn0 10807   VDeg cvdg 24716   RegGrph crgra 24745
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 4574  ax-nul 4582  ax-pr 4692
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-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298  df-oprab 6299  df-rgra 24747
This theorem is referenced by:  isrusgra  24750  rgraprop  24751  rusgrargra  24753  isrusgusrg  24755  isrgrac  24757  0egra0rgra  24759  0vgrargra  24760
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