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Theorem dfnbgr5 47775
Description: Alternate definition of the (open) neighborhood of a vertex as a semiclosed neighborhood without itself. (Contributed by AV, 16-May-2025.)
Hypotheses
Ref Expression
dfsclnbgr2.v 𝑉 = (Vtx‘𝐺)
dfsclnbgr2.s 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
dfsclnbgr2.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
dfnbgr5 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
Distinct variable groups:   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑒,𝐸,𝑛   𝑒,𝐺,𝑛
Allowed substitution hints:   𝑆(𝑒,𝑛)

Proof of Theorem dfnbgr5
StepHypRef Expression
1 rabdif 4327 . . 3 ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)}
21eqcomi 2744 . 2 {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} = ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁})
3 dfsclnbgr2.v . . 3 𝑉 = (Vtx‘𝐺)
4 dfsclnbgr2.e . . 3 𝐸 = (Edg‘𝐺)
53, 4dfnbgr2 29369 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
6 dfsclnbgr2.s . . . 4 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
73, 6, 4dfsclnbgr2 47770 . . 3 (𝑁𝑉𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
87difeq1d 4135 . 2 (𝑁𝑉 → (𝑆 ∖ {𝑁}) = ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}))
92, 5, 83eqtr4a 2801 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  cdif 3960  wss 3963  {csn 4631  {cpr 4633  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079   NeighbVtx cnbgr 29364
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-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-oprab 7435  df-mpo 7436  df-nbgr 29365
This theorem is referenced by:  dfnbgrss  47776
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