Theorem dfnbgr5 47837

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 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
dfsclnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
dfnbgr5	⊢ (𝑁 ∈ 𝑉 → (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

Step	Hyp	Ref	Expression
1		rabdif 4321	. . 3 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
2	1	eqcomi 2746	. 2 ⊢ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})
3		dfsclnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
4		dfsclnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
5	3, 4	dfnbgr2 29354	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
6		dfsclnbgr2.s	. . . 4 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
7	3, 6, 4	dfsclnbgr2 47832	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
8	7	difeq1d 4125	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑆 ∖ {𝑁}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}))
9	2, 5, 8	3eqtr4a 2803	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ⊆ wss 3951  {csn 4626  {cpr 4628  ‘cfv 6561  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-nbgr 29350

This theorem is referenced by:  dfnbgrss  47838