Theorem rusgrprop 29763

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.

Step	Hyp	Ref	Expression
1		df-rusgr 29759	. . 3 ⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
2	1	bropaex12 5738	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3		isrusgr 29762	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
4	3	biimpd 231	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
5	2, 4	mpcom 38	1 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100  USGraphcusgr 29350   RegGraph crgr 29756   RegUSGraph crusgr 29757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rusgr 29759

This theorem is referenced by:  rusgrrgr  29764  rusgrusgr  29765  rusgrprop0  29768