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

Theorem isrgra 26453
 Description: The property of being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
Assertion
Ref Expression
isrgra ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ 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 isrgra
Dummy variables 𝑒 𝑘 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4584 . 2 (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ RegGrph )
2 df-rgra 26451 . . . 4 RegGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}
32eleq2i 2680 . . 3 (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ RegGrph ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)})
4 eleq1 2676 . . . . . 6 (𝑘 = 𝐾 → (𝑘 ∈ ℕ0𝐾 ∈ ℕ0))
543ad2ant3 1077 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → (𝑘 ∈ ℕ0𝐾 ∈ ℕ0))
6 simp1 1054 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → 𝑣 = 𝑉)
7 oveq12 6558 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 VDeg 𝑒) = (𝑉 VDeg 𝐸))
87fveq1d 6105 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑣 VDeg 𝑒)‘𝑝) = ((𝑉 VDeg 𝐸)‘𝑝))
983adant3 1074 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → ((𝑣 VDeg 𝑒)‘𝑝) = ((𝑉 VDeg 𝐸)‘𝑝))
10 simp3 1056 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → 𝑘 = 𝐾)
119, 10eqeq12d 2625 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → (((𝑣 VDeg 𝑒)‘𝑝) = 𝑘 ↔ ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
126, 11raleqbidv 3129 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → (∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘 ↔ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
135, 12anbi12d 743 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸𝑘 = 𝐾) → ((𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘) ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
1413eloprabga 6645 . . 3 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)} ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
153, 14syl5bb 271 . 2 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ RegGrph ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
161, 15syl5bb 271 1 ((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ 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   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-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 This theorem is referenced by:  isrusgra  26454  rgraprop  26455  rusgrargra  26457  isrusgusrg  26459  isrgrac  26461  0egra0rgra  26463  0vgrargra  26464
 Copyright terms: Public domain W3C validator