Theorem uvtxaisvtx 40615

Description: A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)

Hypothesis

Ref	Expression
uvtxael.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxaisvtx	⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ 𝑉)

Proof of Theorem uvtxaisvtx

Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6131	. 2 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝐺 ∈ V)
2		uvtxael.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	uvtxaval 40613	. . . 4 ⊢ (𝐺 ∈ V → (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
4	3	eleq2d 2673	. . 3 ⊢ (𝐺 ∈ V → (𝑁 ∈ (UnivVtx‘𝐺) ↔ 𝑁 ∈ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}))
5		elrabi 3328	. . 3 ⊢ (𝑁 ∈ {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} → 𝑁 ∈ 𝑉)
6	4, 5	syl6bi 242	. 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ 𝑉))
7	1, 6	mpcom 37	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537  {csn 4125  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   NeighbVtx cnbgr 40550  UnivVtxcuvtxa 40551

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-8 1979  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-pow 4769  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-csb 3500  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-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-uvtxa 40556

This theorem is referenced by:  uvtxassvtx  40616  uvtxanm1nbgr  40631  cplgruvtxb  40637  vdiscusgr  40747