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

Theorem finrusgrfusgr 26338
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 26337 . . 3 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )
21anim1i 591 . 2 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
3 finrusgrfusgr.v . . 3 𝑉 = (Vtx‘𝐺)
43isfusgr 26105 . 2 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
52, 4sylibr 224 1 ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   class class class wbr 4615  cfv 5849  Fincfn 7902  Vtxcvtx 25781   USGraph cusgr 25944   FinUSGraph cfusgr 26103   RegUSGraph crusgr 26329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-iota 5812  df-fv 5857  df-fusgr 26104  df-rusgr 26331
This theorem is referenced by:  numclwwlk1  27093  numclwwlk3  27104  numclwwlk5  27107  numclwwlk7lem  27108  numclwwlk6  27109  frgrreggt1  27112
  Copyright terms: Public domain W3C validator