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

Theorem isrusgr 27649
Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.)
Assertion
Ref Expression
isrusgr ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))

Proof of Theorem isrusgr
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2825 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
21adantr 484 . . 3 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
3 breq12 5058 . . 3 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑔 RegGraph 𝑘𝐺 RegGraph 𝐾))
42, 3anbi12d 634 . 2 ((𝑔 = 𝐺𝑘 = 𝐾) → ((𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘) ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
5 df-rusgr 27646 . 2 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
64, 5brabga 5415 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110   class class class wbr 5053  USGraphcusgr 27240   RegGraph crgr 27643   RegUSGraph crusgr 27644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-rusgr 27646
This theorem is referenced by:  rusgrprop  27650  isrusgr0  27654  usgr0edg0rusgr  27663  0vtxrusgr  27665
  Copyright terms: Public domain W3C validator