MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  finrusgrfusgr Structured version   Visualization version   GIF version

Theorem finrusgrfusgr 26815
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 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)

Proof of Theorem finrusgrfusgr
StepHypRef Expression
1 rusgrusgr 26814 . . 3 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
21anim1i 609 . 2 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 finrusgrfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
43isfusgr 26552 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
52, 4sylibr 226 1 ((𝐺RegUSGraph𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157   class class class wbr 4843  cfv 6101  Fincfn 8195  Vtxcvtx 26231  USGraphcusgr 26385  FinUSGraphcfusgr 26550  RegUSGraphcrusgr 26806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-iota 6064  df-fv 6109  df-fusgr 26551  df-rusgr 26808
This theorem is referenced by:  numclwwlk1  27729  numclwwlk3  27770  numclwwlk5  27773  numclwwlk7lem  27774  numclwwlk6  27775  frgrreggt1  27778
  Copyright terms: Public domain W3C validator