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Theorem uvtxel1 29428
Description: Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 14-Feb-2022.)
Hypotheses
Ref Expression
uvtxel.v 𝑉 = (Vtx‘𝐺)
isuvtx.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
uvtxel1 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑘   𝑒,𝑉,𝑘   𝑒,𝑁,𝑘
Allowed substitution hint:   𝐸(𝑘)

Proof of Theorem uvtxel1
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 sneq 4641 . . . 4 (𝑛 = 𝑁 → {𝑛} = {𝑁})
21difeq2d 4136 . . 3 (𝑛 = 𝑁 → (𝑉 ∖ {𝑛}) = (𝑉 ∖ {𝑁}))
3 preq2 4739 . . . . 5 (𝑛 = 𝑁 → {𝑘, 𝑛} = {𝑘, 𝑁})
43sseq1d 4027 . . . 4 (𝑛 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑘, 𝑁} ⊆ 𝑒))
54rexbidv 3177 . . 3 (𝑛 = 𝑁 → (∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
62, 5raleqbidv 3344 . 2 (𝑛 = 𝑁 → (∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒 ↔ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
7 uvtxel.v . . 3 𝑉 = (Vtx‘𝐺)
8 isuvtx.e . . 3 𝐸 = (Edg‘𝐺)
97, 8isuvtx 29427 . 2 (UnivVtx‘𝐺) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛})∃𝑒𝐸 {𝑘, 𝑛} ⊆ 𝑒}
106, 9elrab2 3698 1 (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cdif 3960  wss 3963  {csn 4631  {cpr 4633  cfv 6563  Vtxcvtx 29028  Edgcedg 29079  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  ax-un 7754
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-sbc 3792  df-csb 3909  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-iun 4998  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-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-nbgr 29365  df-uvtx 29418
This theorem is referenced by: (None)
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