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Theorem dfclnbgr4 47816
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 2736 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2dfclnbgr2 47815 . 2 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}))
4 undif2 4476 . . . 4 ({𝑁} ∪ ({𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
5 rabdif 4320 . . . . 5 ({𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}
65uneq2i 4164 . . . 4 ({𝑁} ∪ ({𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
74, 6eqtr3i 2766 . . 3 ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
81, 2dfnbgr2 29355 . . . . 5 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)})
98eqcomd 2742 . . . 4 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)} = (𝐺 NeighbVtx 𝑁))
109uneq2d 4167 . . 3 (𝑁𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
117, 10eqtrid 2788 . 2 (𝑁𝑉 → ({𝑁} ∪ {𝑛𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁𝑒𝑛𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
123, 11eqtrd 2776 1 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wrex 3069  {crab 3435  cdif 3947  cun 3948  {csn 4625  cfv 6560  (class class class)co 7432  Vtxcvtx 29014  Edgcedg 29065   NeighbVtx cnbgr 29350   ClNeighbVtx cclnbgr 47810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-nbgr 29351  df-clnbgr 47811
This theorem is referenced by:  elclnbgrelnbgr  47817  clnbupgr  47825  clnbgr0edg  47828  edgusgrclnbfin  47833  stgrclnbgr0  47937  isubgr3stgrlem1  47938
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