Theorem rusgrprop 29598

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 29594	. . 3 ⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
2	1	bropaex12 5791	. 2 ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3		isrusgr 29597	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
4	3	biimpd 229	. 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 395   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166  USGraphcusgr 29184   RegGraph crgr 29591   RegUSGraph crusgr 29592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rusgr 29594

This theorem is referenced by:  rusgrrgr  29599  rusgrusgr  29600  rusgrprop0  29603