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Theorem isrusgusrg 26459
 Description: A graph is a k-regular undirected simple graph iff it is an undirected simple graph and a k-regular graph. (Contributed by AV, 3-Jan-2020.)
Assertion
Ref Expression
isrusgusrg ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))

Proof of Theorem isrusgusrg
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 isrusgra 26454 . 2 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
2 3anass 1035 . . 3 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾) ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
3 isrgra 26453 . . . . 5 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
43bicomd 212 . . . 4 ((𝑉𝑋𝐸𝑌𝐾𝑍) → ((𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾) ↔ ⟨𝑉, 𝐸⟩ RegGrph 𝐾))
54anbi2d 736 . . 3 ((𝑉𝑋𝐸𝑌𝐾𝑍) → ((𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)) ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
62, 5syl5bb 271 . 2 ((𝑉𝑋𝐸𝑌𝐾𝑍) → ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾) ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
71, 6bitrd 267 1 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℕ0cn0 11169   USGrph cusg 25859   VDeg cvdg 26420   RegGrph crgra 26449   RegUSGrph crusgra 26450 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  isrusgusrgcl  26460
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