Theorem uvtxval 27754

Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (Revised by AV, 14-Feb-2022.)

Hypothesis

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

Assertion

Ref	Expression
uvtxval	⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}

Distinct variable groups:   𝑛,𝐺,𝑣   𝑛,𝑉,𝑣

Proof of Theorem uvtxval

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-uvtx 27753	. . 3 ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
2		fveq2 6774	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3		uvtxval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4	2, 3	eqtr4di 2796	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
5	4	difeq1d 4056	. . . 4 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
6		oveq1 7282	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
7	6	eleq2d 2824	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8	5, 7	raleqbidv 3336	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
9	1, 8	fvmptrabfv 6906	. 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
10	3	eqcomi 2747	. . 3 ⊢ (Vtx‘𝐺) = 𝑉
11	10	rabeqi 3416	. 2 ⊢ {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
12	9, 11	eqtri 2766	1 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}

Syntax hints:   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ∖ cdif 3884  {csn 4561  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366   NeighbVtx cnbgr 27699  UnivVtxcuvtx 27752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-uvtx 27753

This theorem is referenced by:  uvtxel  27755  uvtx0  27761  isuvtx  27762  uvtx01vtx  27764  uvtxusgr  27769