Theorem rusisusgra 26458

Description: Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)

Assertion

Ref	Expression
rusisusgra	⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → 𝑉 USGrph 𝐸)

Proof of Theorem rusisusgra

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rusgraprop 26456	. 2 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
2	1	simp1d 1066	1 ⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → 𝑉 USGrph 𝐸)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℕ₀cn0 11169   USGrph cusg 25859   VDeg cvdg 26420   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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-opab 4644  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-rgra 26451  df-rusgra 26452

This theorem is referenced by:  rusgranumwwlkl1  26473  rusgranumwlkl1  26474  rusgranumwlkb1  26481  rusgra0edg  26482  rusgranumwlks  26483  rusgranumwlk  26484  rusgranumwwlkg  26486  numclwwlkovf2num  26612  numclwwlk1  26625  numclwwlkqhash  26627  numclwwlk3  26636  numclwwlk5  26639  numclwwlk6  26640  frgrareg  26644