Theorem dfclnbgr4 48012

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	⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Proof of Theorem dfclnbgr4

Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclnbgr4.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2734	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	dfclnbgr2 48011	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
4		undif2 4427	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
5		rabdif 4271	. . . . 5 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
6	5	uneq2i 4115	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
7	4, 6	eqtr3i 2759	. . 3 ⊢ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
8	1, 2	dfnbgr2 29359	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	eqcomd 2740	. . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = (𝐺 NeighbVtx 𝑁))
10	9	uneq2d 4118	. . 3 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
11	7, 10	eqtrid 2781	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
12	3, 11	eqtrd 2769	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397   ∖ cdif 3896   ∪ cun 3897  {csn 4578  ‘cfv 6490  (class class class)co 7356  Vtxcvtx 29018  Edgcedg 29069   NeighbVtx cnbgr 29354   ClNeighbVtx cclnbgr 48006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-nbgr 29355  df-clnbgr 48007

This theorem is referenced by:  elclnbgrelnbgr  48013  clnbupgr  48021  clnbgr0edg  48025  edgusgrclnbfin  48030  stgrclnbgr0  48153  isubgr3stgrlem1  48154