Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isrusgra Structured version   Visualization version   GIF version

Theorem isrusgra 26454
 Description: The property of being a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
Assertion
Ref Expression
isrusgra ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
Distinct variable groups:   𝐸,𝑝   𝐾,𝑝   𝑉,𝑝
Allowed substitution hints:   𝑋(𝑝)   𝑌(𝑝)   𝑍(𝑝)

Proof of Theorem isrusgra
Dummy variables 𝑒 𝑘 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4584 . . . . 5 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ RegUSGrph )
2 df-rusgra 26452 . . . . . 6 RegUSGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)}
32eleq2i 2680 . . . . 5 (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ RegUSGrph ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)})
41, 3bitri 263 . . . 4 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)})
5 breq12 4588 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 USGrph 𝑒𝑉 USGrph 𝐸))
653adant3 1074 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → (𝑣 USGrph 𝑒𝑉 USGrph 𝐸))
7 opeq12 4342 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → ⟨𝑣, 𝑒⟩ = ⟨𝑉, 𝐸⟩)
873adant3 1074 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → ⟨𝑣, 𝑒⟩ = ⟨𝑉, 𝐸⟩)
9 simp3 1056 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → 𝑘 = 𝐾)
108, 9breq12d 4596 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → (⟨𝑣, 𝑒⟩ RegGrph 𝑘 ↔ ⟨𝑉, 𝐸⟩ RegGrph 𝐾))
116, 10anbi12d 743 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → ((𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘) ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
1211eloprabga 6645 . . . 4 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)} ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
134, 12syl5bb 271 . . 3 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
14 isrgra 26453 . . . 4 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
1514anbi2d 736 . . 3 ((𝑉𝑋𝐸𝑌𝐾𝑍) → ((𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾) ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))))
1613, 15bitrd 267 . 2 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))))
17 3anass 1035 . 2 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾) ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
1816, 17syl6bbr 277 1 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 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  {coprab 6550  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:  rusgraprop  26456  rusgrargra  26457  isrusgusrg  26459  cusgraisrusgra  26465  frgraregorufrg  26599
 Copyright terms: Public domain W3C validator