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Theorem rusgraprop 30686
 Description: The properties of a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rusgraprop RegUSGrph USGrph VDeg
Distinct variable groups:   ,   ,   ,

Proof of Theorem rusgraprop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgra 30682 . . . 4 RegUSGrph USGrph RegGrph
21breqi 4398 . . 3 RegUSGrph USGrph RegGrph
3 oprabv 30297 . . 3 USGrph RegGrph
42, 3sylbi 195 . 2 RegUSGrph
5 isrusgra 30684 . . 3 RegUSGrph USGrph VDeg
65biimpd 207 . 2 RegUSGrph USGrph VDeg
74, 6mpcom 36 1 RegUSGrph USGrph VDeg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 965   wceq 1370   wcel 1758  wral 2795  cvv 3070  cop 3983   class class class wbr 4392  cfv 5518  (class class class)co 6192  coprab 6193  cn0 10682   USGrph cusg 23401   VDeg cvdg 23700   RegGrph crgra 30679   RegUSGrph crusgra 30680 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-iota 5481  df-fv 5526  df-ov 6195  df-oprab 6196  df-rgra 30681  df-rusgra 30682 This theorem is referenced by:  rusisusgra  30688  cusgraiffrusgra  30693  rusgraprop2  30694  rusgranumwlks  30714  rusgranumwlk  30715  frrusgraord  30804  numclwwlk3  30842  numclwwlk5lem  30844  numclwwlk5  30845  numclwwlk7  30847  frgrareggt1  30849  frgrareg  30850  frgraregord013  30851
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