Theorem rgrprop 29085

Description: The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)

Hypotheses

Ref	Expression
isrgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isrgr.d	⊢ 𝐷 = (VtxDeg‘𝐺)

Assertion

Ref	Expression
rgrprop	⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Distinct variable groups:   𝑣,𝐺   𝑣,𝐾

Allowed substitution hints:   𝐷(𝑣)   𝑉(𝑣)

Proof of Theorem rgrprop

Dummy variables 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rgr 29082	. . 3 ⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
2	1	bropaex12 5767	. 2 ⊢ (𝐺 RegGraph 𝐾 → (𝐺 ∈ V ∧ 𝐾 ∈ V))
3		isrgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
4		isrgr.d	. . . 4 ⊢ 𝐷 = (VtxDeg‘𝐺)
5	3, 4	isrgr 29084	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
6	5	biimpd 228	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
7	2, 6	mpcom 38	1 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   class class class wbr 5148  ‘cfv 6543  ℕ₀^*cxnn0 12549  Vtxcvtx 28524  VtxDegcvtxdg 28990   RegGraph crgr 29080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-iota 6495  df-fv 6551  df-rgr 29082

This theorem is referenced by:  rusgrprop0  29092  uhgr0edg0rgrb  29099  frrusgrord  29862