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Theorem isrgra 25496
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 4418 . 2  |-  ( <. V ,  E >. RegGrph  K  <->  <. <. V ,  E >. ,  K >.  e. RegGrph  )
2 df-rgra 25494 . . . 4  |- RegGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( k  e.  NN0  /\ 
A. p  e.  v  ( ( v VDeg  e
) `  p )  =  k ) }
32eleq2i 2498 . . 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 2492 . . . . . 6  |-  ( k  =  K  ->  (
k  e.  NN0  <->  K  e.  NN0 ) )
543ad2ant3 1028 . . . . 5  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( k  e.  NN0  <->  K  e.  NN0 ) )
6 simp1 1005 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  v  =  V )
7 oveq12 6305 . . . . . . . . 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 1025 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( v VDeg  e
) `  p )  =  ( ( V VDeg 
E ) `  p
) )
10 simp3 1007 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  k  =  K )
119, 10eqeq12d 2442 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( ( v VDeg  e ) `  p
)  =  k  <->  ( ( V VDeg  E ) `  p
)  =  K ) )
126, 11raleqbidv 3037 . . . . 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 715 . . . 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 6388 . . 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 260 . 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 260 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   <.cop 3999   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   {coprab 6297   NN0cn0 10858   VDeg cvdg 25463   RegGrph crgra 25492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-oprab 6300  df-rgra 25494
This theorem is referenced by:  isrusgra  25497  rgraprop  25498  rusgrargra  25500  isrusgusrg  25502  isrgrac  25504  0egra0rgra  25506  0vgrargra  25507
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