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Theorem isrusgra 24603
Description: The property of being a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
Assertion
Ref Expression
isrusgra  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. 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
Allowed substitution hints:    X( p)    Y( p)    Z( p)

Proof of Theorem isrusgra
Dummy variables  e 
k  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4448 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. <. V ,  E >. ,  K >.  e. RegUSGrph  )
2 df-rusgra 24601 . . . . . 6  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
32eleq2i 2545 . . . . 5  |-  ( <. <. V ,  E >. ,  K >.  e. RegUSGrph  <->  <. <. V ,  E >. ,  K >.  e. 
{ <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) } )
41, 3bitri 249 . . . 4  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. <. V ,  E >. ,  K >.  e.  { <. <.
v ,  e >. ,  k >.  |  ( v USGrph  e  /\  <. v ,  e >. RegGrph  k ) } )
5 breq12 4452 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
653adant3 1016 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( v USGrph  e  <->  V USGrph  E ) )
7 opeq12 4215 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  -> 
<. v ,  e >.  =  <. V ,  E >. )
873adant3 1016 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  -> 
<. v ,  e >.  =  <. V ,  E >. )
9 simp3 998 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  k  =  K )
108, 9breq12d 4460 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( <. v ,  e
>. RegGrph  k  <->  <. V ,  E >. RegGrph 
K ) )
116, 10anbi12d 710 . . . . 5  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( v USGrph  e  /\  <. v ,  e
>. RegGrph  k )  <->  ( V USGrph  E  /\  <. V ,  E >. RegGrph 
K ) ) )
1211eloprabga 6371 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. <. V ,  E >. ,  K >.  e.  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }  <->  ( V USGrph  E  /\  <. V ,  E >. RegGrph 
K ) ) )
134, 12syl5bb 257 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. V ,  E >. RegUSGrph  K 
<->  ( V USGrph  E  /\  <. V ,  E >. RegGrph  K
) ) )
14 isrgra 24602 . . . 4  |-  ( ( 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 ) ) )
1514anbi2d 703 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( ( V USGrph  E  /\  <. V ,  E >. RegGrph 
K )  <->  ( V USGrph  E  /\  ( K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K ) ) ) )
1613, 15bitrd 253 . 2  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. V ,  E >. RegUSGrph  K 
<->  ( V USGrph  E  /\  ( K  e.  NN0  /\ 
A. p  e.  V  ( ( V VDeg  E
) `  p )  =  K ) ) ) )
17 3anass 977 . 2  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. p  e.  V  ( ( V VDeg  E ) `  p
)  =  K )  <-> 
( V USGrph  E  /\  ( K  e.  NN0  /\ 
A. p  e.  V  ( ( V VDeg  E
) `  p )  =  K ) ) )
1816, 17syl6bbr 263 1  |-  ( ( V  e.  X  /\  E  e.  Y  /\  K  e.  Z )  ->  ( <. 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    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   {coprab 6283   NN0cn0 10791   USGrph cusg 24006   VDeg cvdg 24569   RegGrph crgra 24598   RegUSGrph crusgra 24599
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-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-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-iota 5549  df-fv 5594  df-ov 6285  df-oprab 6286  df-rgra 24600  df-rusgra 24601
This theorem is referenced by:  rusgraprop  24605  rusgrargra  24606  isrusgusrg  24608  cusgraisrusgra  24614  frgraregorufrg  24749
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