Theorem isrusgr 30029

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 2850	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
2	1	adantr 486	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
3		breq12 5112	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 RegGraph 𝑘 ↔ 𝐺 RegGraph 𝐾))
4	2, 3	anbi12d 644	. 2 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → ((𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘) ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
5		df-rusgr 30026	. 2 ⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
6	4, 5	brabga 5516	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5107  USGraphcusgr 29617   RegGraph crgr 30023   RegUSGraph crusgr 30024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-rusgr 30026

This theorem is used by:  rusgrprop  30030  isrusgr0  30034  usgr0edg0rusgr  30043  0vtxrusgr  30045