Theorem isrusgr 30148

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 2848	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
2	1	adantr 486	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
3		breq12 5107	. . 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 30145	. 2 ⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
6	4, 5	brabga 5507	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 1569   ∈ wcel 2144   class class class wbr 5102  USGraphcusgr 29736   RegGraph crgr 30142   RegUSGraph crusgr 30143

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-rusgr 30145

This theorem is used by:  rusgrprop  30149  isrusgr0  30153  usgr0edg0rusgr  30162  0vtxrusgr  30164