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Theorem dfnbgr2 27714
Description: Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
nbgrval.v 𝑉 = (Vtx‘𝐺)
nbgrval.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
dfnbgr2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑛   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem dfnbgr2
StepHypRef Expression
1 nbgrval.v . . 3 𝑉 = (Vtx‘𝐺)
2 nbgrval.e . . 3 𝐸 = (Edg‘𝐺)
31, 2nbgrval 27713 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4 prssg 4752 . . . . . 6 ((𝑁𝑉𝑛 ∈ V) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
54elvd 3436 . . . . 5 (𝑁𝑉 → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
65bicomd 222 . . . 4 (𝑁𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
76rexbidv 3224 . . 3 (𝑁𝑉 → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)))
87rabbidv 3411 . 2 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
93, 8eqtrd 2778 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  {crab 3068  Vcvv 3429  cdif 3883  wss 3886  {csn 4561  {cpr 4563  cfv 6426  (class class class)co 7267  Vtxcvtx 27376  Edgcedg 27427   NeighbVtx cnbgr 27709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-nbgr 27710
This theorem is referenced by: (None)
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