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Theorem uvtxel 29423
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) ↔ (𝑁𝑉 ∧ ∀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.
StepHypRef Expression
1 sneq 4658 . . . 4 (𝑣 = 𝑁 → {𝑣} = {𝑁})
21difeq2d 4149 . . 3 (𝑣 = 𝑁 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑁}))
3 oveq2 7456 . . . 4 (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
43eleq2d 2830 . . 3 (𝑣 = 𝑁 → (𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
52, 4raleqbidv 3354 . 2 (𝑣 = 𝑁 → (∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
6 uvtxel.v . . 3 𝑉 = (Vtx‘𝐺)
76uvtxval 29422 . 2 (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
85, 7elrab2 3711 1 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2108  wral 3067  cdif 3973  {csn 4648  cfv 6573  (class class class)co 7448  Vtxcvtx 29031   NeighbVtx cnbgr 29367  UnivVtxcuvtx 29420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-uvtx 29421
This theorem is referenced by:  uvtxisvtx  29424  vtxnbuvtx  29426  uvtx2vtx1edg  29433  uvtx2vtx1edgb  29434  uvtxnbgrb  29436  iscplgrnb  29451  cplgr1v  29465  cusgrexi  29478  structtocusgr  29481
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