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Theorem rusgrprop 29328
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 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))

Proof of Theorem rusgrprop
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rusgr 29324 . . 3 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
21bropaex12 5760 . 2 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3 isrusgr 29327 . . 3 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
43biimpd 228 . 2 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
52, 4mpcom 38 1 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  Vcvv 3468   class class class wbr 5141  USGraphcusgr 28917   RegGraph crgr 29321   RegUSGraph crusgr 29322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rusgr 29324
This theorem is referenced by:  rusgrrgr  29329  rusgrusgr  29330  rusgrprop0  29333
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