Theorem rusgrprop

Description: The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018) (Revised by AV, 18-Dec-2020)

	Ref	Expression
Assertion	rusgrprop	⊢ ( 𝐺 RegUSGraph 𝐾 → ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) )

Proof

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