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Theorem rgraprop 26455
 Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
Assertion
Ref Expression
rgraprop (⟨𝑉, 𝐸⟩ RegGrph 𝐾 → (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
Distinct variable groups:   𝐸,𝑝   𝐾,𝑝   𝑉,𝑝

Proof of Theorem rgraprop
Dummy variables 𝑒 𝑘 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rgra 26451 . . . 4 RegGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}
21breqi 4589 . . 3 (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ ⟨𝑉, 𝐸⟩{⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}𝐾)
3 oprabv 6601 . . 3 (⟨𝑉, 𝐸⟩{⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}𝐾 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝐾 ∈ V))
42, 3sylbi 206 . 2 (⟨𝑉, 𝐸⟩ RegGrph 𝐾 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝐾 ∈ V))
5 isrgra 26453 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝐾 ∈ V) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
65biimpd 218 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝐾 ∈ V) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 → (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
74, 6mpcom 37 1 (⟨𝑉, 𝐸⟩ RegGrph 𝐾 → (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  {coprab 6550  ℕ0cn0 11169   VDeg cvdg 26420   RegGrph crgra 26449 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-rgra 26451 This theorem is referenced by:  0eusgraiff0rgra  26466
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