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.

Step	Hyp	Ref	Expression
1		df-rusgr 27046	. . 3 ⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)}
2	1	bropaex12 5493	. 2 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3		isrusgr 27049	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
4	3	biimpd 221	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
5	2, 4	mpcom 38	1 ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))

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