Theorem isrusgr 29086

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.)

Assertion

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

Proof of Theorem isrusgr

Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2820	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
2	1	adantr 480	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
3		breq12 5153	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 RegGraph 𝑘 ↔ 𝐺 RegGraph 𝐾))
4	2, 3	anbi12d 630	. 2 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘) ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
5		df-rusgr 29083	. 2 ⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
6	4, 5	brabga 5534	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   class class class wbr 5148  USGraphcusgr 28677   RegGraph crgr 29080   RegUSGraph crusgr 29081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-rusgr 29083

This theorem is referenced by:  rusgrprop  29087  isrusgr0  29091  usgr0edg0rusgr  29100  0vtxrusgr  29102