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Theorem rgrprop 27269
Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Hypotheses
Ref Expression
isrgr.v 𝑉 = (Vtx‘𝐺)
isrgr.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
rgrprop (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
Distinct variable groups:   𝑣,𝐺   𝑣,𝐾
Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)

Proof of Theorem rgrprop
Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rgr 27266 . . 3 RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
21bropaex12 5635 . 2 (𝐺 RegGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3 isrgr.v . . . 4 𝑉 = (Vtx‘𝐺)
4 isrgr.d . . . 4 𝐷 = (VtxDeg‘𝐺)
53, 4isrgr 27268 . . 3 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
65biimpd 230 . 2 ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
72, 6mpcom 38 1 (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492   class class class wbr 5057  cfv 6348  0*cxnn0 11955  Vtxcvtx 26708  VtxDegcvtxdg 27174   RegGraph crgr 27264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-iota 6307  df-fv 6356  df-rgr 27266
This theorem is referenced by:  rusgrprop0  27276  uhgr0edg0rgrb  27283  frrusgrord  28047
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