Theorem uvtxel 29457

Description: A universal vertex, i.e. an element of 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
uvtxel.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
uvtxel	⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))

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

Proof of Theorem uvtxel

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4577	. . . 4 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
2	1	difeq2d 4066	. . 3 ⊢ (𝑣 = 𝑁 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑁}))
3		oveq2 7375	. . . 4 ⊢ (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
4	3	eleq2d 2822	. . 3 ⊢ (𝑣 = 𝑁 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
5	2, 4	raleqbidv 3311	. 2 ⊢ (𝑣 = 𝑁 → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
6		uvtxel.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
7	6	uvtxval 29456	. 2 ⊢ (UnivVtx‘𝐺) = {𝑣 ∈ 𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
8	5, 7	elrab2 3637	1 ⊢ (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ∖ cdif 3886  {csn 4567  ‘cfv 6498  (class class class)co 7367  Vtxcvtx 29065   NeighbVtx cnbgr 29401  UnivVtxcuvtx 29454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-uvtx 29455

This theorem is referenced by:  uvtxisvtx  29458  vtxnbuvtx  29460  uvtx2vtx1edg  29467  uvtx2vtx1edgb  29468  uvtxnbgrb  29470  iscplgrnb  29485  cplgr1v  29499  cusgrexi  29512  structtocusgr  29515