Theorem uvtxval 27171

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 27170	. . 3 ⊢ UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
2		fveq2 6672	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3		uvtxval.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4	2, 3	syl6eqr 2876	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
5	4	difeq1d 4100	. . . 4 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
6		oveq1 7165	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
7	6	eleq2d 2900	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
8	5, 7	raleqbidv 3403	. . 3 ⊢ (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
9	1, 8	fvmptrabfv 6801	. 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
10	3	eqcomi 2832	. . 3 ⊢ (Vtx‘𝐺) = 𝑉
11	10	rabeqi 3484	. 2 ⊢ {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
12	9, 11	eqtri 2846	1 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}

Syntax hints:   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144   ∖ cdif 3935  {csn 4569  ‘cfv 6357  (class class class)co 7158  Vtxcvtx 26783   NeighbVtx cnbgr 27116  UnivVtxcuvtx 27169

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-uvtx 27170

This theorem is referenced by:  uvtxel  27172  uvtx0  27178  isuvtx  27179  uvtx01vtx  27181  uvtxusgr  27186