Theorem rgrprop 29851

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 29848	. . 3 ⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
2	1	bropaex12 5753	. 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 29850	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
6	5	biimpd 232	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐾 ∈ V) → (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
7	2, 6	mpcom 39	1 ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   class class class wbr 5113  ‘cfv 6537  ℕ₀^*cxnn0 12577  Vtxcvtx 29287  VtxDegcvtxdg 29756   RegGraph crgr 29846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-iota 6493  df-fv 6545  df-rgr 29848

This theorem is referenced by:  rusgrprop0  29858  uhgr0edg0rgrb  29865  frrusgrord  30633