Theorem isrusgr

Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018) (Revised by AV, 18-Dec-2020)

	Ref	Expression
Assertion	isrusgr	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( 𝐺 RegUSGraph 𝐾 ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eleq1	⊢ ( 𝑔 = 𝐺 → ( 𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph ) )
2	1	adantr	⊢ ( ( 𝑔 = 𝐺 ∧ 𝑘 = 𝐾 ) → ( 𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph ) )
3		breq12	⊢ ( ( 𝑔 = 𝐺 ∧ 𝑘 = 𝐾 ) → ( 𝑔 RegGraph 𝑘 ↔ 𝐺 RegGraph 𝐾 ) )
4	2 3	anbi12d	⊢ ( ( 𝑔 = 𝐺 ∧ 𝑘 = 𝐾 ) → ( ( 𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘 ) ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) ) )
5		df-rusgr	⊢ RegUSGraph = { ⟨ 𝑔 , 𝑘 ⟩ ∣ ( 𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘 ) }
6	4 5	brabga	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍 ) → ( 𝐺 RegUSGraph 𝐾 ↔ ( 𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾 ) ) )