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Theorem isrusgr 28509
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 481 . . 3 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
3 breq12 5110 . . 3 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑔 RegGraph 𝑘𝐺 RegGraph 𝐾))
42, 3anbi12d 631 . 2 ((𝑔 = 𝐺𝑘 = 𝐾) → ((𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘) ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
5 df-rusgr 28506 . 2 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
64, 5brabga 5491 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5105  USGraphcusgr 28100   RegGraph crgr 28503   RegUSGraph crusgr 28504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-rusgr 28506
This theorem is referenced by:  rusgrprop  28510  isrusgr0  28514  usgr0edg0rusgr  28523  0vtxrusgr  28525
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