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Theorem finrusgrfusgr 40765
 Description: A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)
Hypothesis
Ref Expression
finrusgrfusgr.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
finrusgrfusgr ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )

Proof of Theorem finrusgrfusgr
StepHypRef Expression
1 rusgrusgr 40764 . . 3 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )
21anim1i 590 . 2 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 finrusgrfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
43isfusgr 40537 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
52, 4sylibr 223 1 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Fincfn 7841  Vtxcvtx 25673   USGraph cusgr 40379   FinUSGraph cfusgr 40535   RegUSGraph crusgr 40756 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-fusgr 40536  df-rusgr 40758 This theorem is referenced by:  av-numclwwlk1  41528  av-numclwwlk3  41539  av-numclwwlk5  41542  av-numclwwlk7lem  41543  av-numclwwlk6  41544  av-frgrareggt1  41547
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