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Theorem isrusgra 25331
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 4395 . . . . 5  |-  ( <. V ,  E >. RegUSGrph  K  <->  <. <. V ,  E >. ,  K >.  e. RegUSGrph  )
2 df-rusgra 25329 . . . . . 6  |- RegUSGrph  =  { <. <. v ,  e
>. ,  k >.  |  ( v USGrph  e  /\  <.
v ,  e >. RegGrph  k ) }
32eleq2i 2480 . . . . 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 4399 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v USGrph  e  <->  V USGrph  E ) )
653adant3 1017 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( v USGrph  e  <->  V USGrph  E ) )
7 opeq12 4160 . . . . . . . 8  |-  ( ( v  =  V  /\  e  =  E )  -> 
<. v ,  e >.  =  <. V ,  E >. )
873adant3 1017 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  -> 
<. v ,  e >.  =  <. V ,  E >. )
9 simp3 999 . . . . . . 7  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  k  =  K )
108, 9breq12d 4407 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( <. v ,  e
>. RegGrph  k  <->  <. V ,  E >. RegGrph 
K ) )
116, 10anbi12d 709 . . . . 5  |-  ( ( v  =  V  /\  e  =  E  /\  k  =  K )  ->  ( ( v USGrph  e  /\  <. v ,  e
>. RegGrph  k )  <->  ( V USGrph  E  /\  <. V ,  E >. RegGrph 
K ) ) )
1211eloprabga 6369 . . . 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 25330 . . . 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 702 . . 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 978 . 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   <.cop 3977   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   {coprab 6278   NN0cn0 10835   USGrph cusg 24734   VDeg cvdg 25297   RegGrph crgra 25326   RegUSGrph crusgra 25327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-oprab 6281  df-rgra 25328  df-rusgra 25329
This theorem is referenced by:  rusgraprop  25333  rusgrargra  25334  isrusgusrg  25336  cusgraisrusgra  25342  frgraregorufrg  25476
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