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Theorem rusgrprop 26514
Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
Assertion
Ref Expression
rusgrprop (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))

Proof of Theorem rusgrprop
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgr 26510 . . . 4 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)}
21breqi 4691 . . 3 (𝐺RegUSGraph𝐾𝐺{⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)}𝐾)
3 brabv 6741 . . 3 (𝐺{⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)}𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
42, 3sylbi 207 . 2 (𝐺RegUSGraph𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
5 isrusgr 26513 . . 3 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
65biimpd 219 . 2 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
74, 6mpcom 38 1 (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  Vcvv 3231   class class class wbr 4685  {copab 4745  USGraphcusgr 26089  RegGraphcrgr 26507  RegUSGraphcrusgr 26508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-rusgr 26510
This theorem is referenced by:  rusgrrgr  26515  rusgrusgr  26516  rusgrprop0  26519
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