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Theorem dfclnbgr4 47749
Description: Alternate definition of the closed neighborhood of a vertex as union of the vertex with its open neighborhood. (Contributed by AV, 8-May-2025.)
Hypothesis
Ref Expression
dfclnbgr4.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
dfclnbgr4 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Proof of Theorem dfclnbgr4
Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfclnbgr4.v . . 3 𝑉 = (Vtx‘𝐺)
2 eqid 2735 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2dfclnbgr2 47748 . 2 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}))
4 undif2 4483 . . . 4 ({𝑁} ∪ ({𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
5 rabdif 4327 . . . . 5 ({𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}
65uneq2i 4175 . . . 4 ({𝑁} ∪ ({𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
74, 6eqtr3i 2765 . . 3 ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
81, 2dfnbgr2 29369 . . . . 5 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
98eqcomd 2741 . . . 4 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} = (𝐺 NeighbVtx 𝑁))
109uneq2d 4178 . . 3 (𝑁𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
117, 10eqtrid 2787 . 2 (𝑁𝑉 → ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
123, 11eqtrd 2775 1 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  cdif 3960  cun 3961  {csn 4631  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079   NeighbVtx cnbgr 29364   ClNeighbVtx cclnbgr 47743
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-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  df-clnbgr 47744
This theorem is referenced by:  elclnbgrelnbgr  47750  clnbupgr  47758  clnbgr0edg  47761  edgusgrclnbfin  47766  stgrclnbgr0  47868  isubgr3stgrlem1  47869
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