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Theorem rusgraprop 25346
 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 25342 . . . 4 RegUSGrph USGrph RegGrph
21breqi 4401 . . 3 RegUSGrph USGrph RegGrph
3 oprabv 6326 . . 3 USGrph RegGrph
42, 3sylbi 195 . 2 RegUSGrph
5 isrusgra 25344 . . 3 RegUSGrph USGrph VDeg
65biimpd 207 . 2 RegUSGrph USGrph VDeg
74, 6mpcom 34 1 RegUSGrph USGrph VDeg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842  wral 2754  cvv 3059  cop 3978   class class class wbr 4395  cfv 5569  (class class class)co 6278  coprab 6279  cn0 10836   USGrph cusg 24747   VDeg cvdg 25310   RegGrph crgra 25339   RegUSGrph crusgra 25340 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 4517  ax-nul 4525  ax-pr 4630 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-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-iota 5533  df-fv 5577  df-ov 6281  df-oprab 6282  df-rgra 25341  df-rusgra 25342 This theorem is referenced by:  rusisusgra  25348  cusgraiffrusgra  25357  rusgraprop2  25359  rusgranumwlks  25373  rusgranumwlk  25374  frrusgraord  25488  numclwwlk3  25526  numclwwlk5lem  25528  numclwwlk5  25529  numclwwlk7  25531  frgrareggt1  25533  frgrareg  25534  frgraregord013  25535
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