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Theorem isrusgr 26691
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 ((𝐺𝑊𝐾𝑍) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))

Proof of Theorem isrusgr
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2837 . . . . 5 (𝑔 = 𝐺 → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
21adantr 466 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑔 ∈ USGraph ↔ 𝐺 ∈ USGraph))
3 breq12 4789 . . . 4 ((𝑔 = 𝐺𝑘 = 𝐾) → (𝑔RegGraph𝑘𝐺RegGraph𝐾))
42, 3anbi12d 608 . . 3 ((𝑔 = 𝐺𝑘 = 𝐾) → ((𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘) ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
5 df-rusgr 26688 . . 3 RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔RegGraph𝑘)}
64, 5brabga 5122 . 2 ((𝐺𝑊𝐾𝑍) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
7 biidd 252 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾) ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
86, 7bitrd 268 1 ((𝐺𝑊𝐾𝑍) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144   class class class wbr 4784  USGraphcusgr 26265  RegGraphcrgr 26685  RegUSGraphcrusgr 26686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-rusgr 26688
This theorem is referenced by:  rusgrprop  26692  isrusgr0  26696  usgr0edg0rusgr  26705  0vtxrusgr  26707
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