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Theorem uvtxval 29419
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) = {𝑣𝑉 ∣ ∀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.
StepHypRef Expression
1 df-uvtx 29418 . . 3 UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})
2 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
3 uvtxval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
42, 3eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
54difeq1d 4135 . . . 4 (𝑔 = 𝐺 → ((Vtx‘𝑔) ∖ {𝑣}) = (𝑉 ∖ {𝑣}))
6 oveq1 7438 . . . . 5 (𝑔 = 𝐺 → (𝑔 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑣))
76eleq2d 2825 . . . 4 (𝑔 = 𝐺 → (𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ 𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
85, 7raleqbidv 3344 . . 3 (𝑔 = 𝐺 → (∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
91, 8fvmptrabfv 7048 . 2 (UnivVtx‘𝐺) = {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
103eqcomi 2744 . . 3 (Vtx‘𝐺) = 𝑉
1110rabeqi 3447 . 2 {𝑣 ∈ (Vtx‘𝐺) ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
129, 11eqtri 2763 1 (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  {crab 3433  cdif 3960  {csn 4631  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   NeighbVtx cnbgr 29364  UnivVtxcuvtx 29417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-uvtx 29418
This theorem is referenced by:  uvtxel  29420  uvtx0  29426  isuvtx  29427  uvtx01vtx  29429  uvtxusgr  29434
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