Theorem dfclnbgr4 47822

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 2729	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	dfclnbgr2 47821	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
4		undif2 4440	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
5		rabdif 4284	. . . . 5 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
6	5	uneq2i 4128	. . . 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 2754	. . 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 29264	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	eqcomd 2735	. . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = (𝐺 NeighbVtx 𝑁))
10	9	uneq2d 4131	. . 3 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
11	7, 10	eqtrid 2776	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
12	3, 11	eqtrd 2764	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3405   ∖ cdif 3911   ∪ cun 3912  {csn 4589  ‘cfv 6511  (class class class)co 7387  Vtxcvtx 28923  Edgcedg 28974   NeighbVtx cnbgr 29259   ClNeighbVtx cclnbgr 47816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-nbgr 29260  df-clnbgr 47817

This theorem is referenced by:  elclnbgrelnbgr  47823  clnbupgr  47831  clnbgr0edg  47834  edgusgrclnbfin  47839  stgrclnbgr0  47961  isubgr3stgrlem1  47962