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Theorem rusgrprop 27050
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 27046 . . 3 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)}
21bropaex12 5493 . 2 (𝐺RegUSGraph𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3 isrusgr 27049 . . 3 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
43biimpd 221 . 2 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
52, 4mpcom 38 1 (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2050  Vcvv 3415   class class class wbr 4930  USGraphcusgr 26640  RegGraphcrgr 27043  RegUSGraphcrusgr 27044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-br 4931  df-opab 4993  df-xp 5414  df-rusgr 27046
This theorem is referenced by:  rusgrrgr  27051  rusgrusgr  27052  rusgrprop0  27055
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