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Theorem isrusgr0 28512
Description: The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrusgr0.v 𝑉 = (Vtx‘𝐺)
isrusgr0.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
isrusgr0 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)   𝑊(𝑣)   𝑍(𝑣)

Proof of Theorem isrusgr0
StepHypRef Expression
1 isrusgr 28507 . 2 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
2 isrusgr0.v . . . . 5 𝑉 = (Vtx‘𝐺)
3 isrusgr0.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
42, 3isrgr 28505 . . . 4 ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
54anbi2d 629 . . 3 ((𝐺𝑊𝐾𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))))
6 3anass 1095 . . 3 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) ↔ (𝐺 ∈ USGraph ∧ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
75, 6bitr4di 288 . 2 ((𝐺𝑊𝐾𝑍) → ((𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾) ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
81, 7bitrd 278 1 ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105  cfv 6496  0*cxnn0 12484  Vtxcvtx 27945  USGraphcusgr 28098  VtxDegcvtxdg 28411   RegGraph crgr 28501   RegUSGraph crusgr 28502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-iota 6448  df-fv 6504  df-rgr 28503  df-rusgr 28504
This theorem is referenced by:  usgreqdrusgr  28514  cusgrrusgr  28527  rgrusgrprc  28535  rusgrprc  28536
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