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Theorem rusgrprop 27352
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 27348 . . 3 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
21bropaex12 5606 . 2 (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3 isrusgr 27351 . . 3 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
43biimpd 232 . 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 399  wcel 2111  Vcvv 3441   class class class wbr 5030  USGraphcusgr 26942   RegGraph crgr 27345   RegUSGraph crusgr 27346
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rusgr 27348
This theorem is referenced by:  rusgrrgr  27353  rusgrusgr  27354  rusgrprop0  27357
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