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	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((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.

Step	Hyp	Ref	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)}
3	2	eleq2i 2680	. . . . 5 ⊢ (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ RegUSGrph ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)})
4	1, 3	bitri 263	. . . 4 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ ⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)})
5		breq12 4588	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣 USGrph 𝑒 ↔ 𝑉 USGrph 𝐸))
6	5	3adant3 1074	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸 ∧ 𝑘 = 𝐾) → (𝑣 USGrph 𝑒 ↔ 𝑉 USGrph 𝐸))
7		opeq12 4342	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ⟨𝑣, 𝑒⟩ = ⟨𝑉, 𝐸⟩)
8	7	3adant3 1074	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸 ∧ 𝑘 = 𝐾) → ⟨𝑣, 𝑒⟩ = ⟨𝑉, 𝐸⟩)
9		simp3 1056	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾)
10	8, 9	breq12d 4596	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸 ∧ 𝑘 = 𝐾) → (⟨𝑣, 𝑒⟩ RegGrph 𝑘 ↔ ⟨𝑉, 𝐸⟩ RegGrph 𝐾))
11	6, 10	anbi12d 743	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸 ∧ 𝑘 = 𝐾) → ((𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘) ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
12	11	eloprabga 6645	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨⟨𝑉, 𝐸⟩, 𝐾⟩ ∈ {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)} ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
13	4, 12	syl5bb 271	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
14		isrgra 26453	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
15	14	anbi2d 736	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → ((𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾) ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))))
16	13, 15	bitrd 267	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))))
17		3anass 1035	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾) ↔ (𝑉 USGrph 𝐸 ∧ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
18	16, 17	syl6bbr 277	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))

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