Theorem dfsclnbgr2 48034

Description: Alternate definition of the semiclosed neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiclosed neighborhood 𝑆 of a vertex 𝑁 is the set of all vertices incident with edges which join the vertex 𝑁 with a vertex. Therefore, a vertex is contained in its semiclosed neighborhood if it is connected with any vertex by an edge (see sclnbgrelself 48036), even only with itself (i.e., by a loop). (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
dfsclnbgr2	⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})

Distinct variable groups:   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛

Allowed substitution hints:   𝑆(𝑒,𝑛)   𝐸(𝑒,𝑛)   𝐺(𝑒,𝑛)

Proof of Theorem dfsclnbgr2

Step	Hyp	Ref	Expression
1		dfsclnbgr2.s	. 2 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
2		prssg 4773	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
3	2	bicomd 223	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
4	3	rexbidv 3158	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
5	4	rabbidva 3403	. 2 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
6	1, 5	eqtrid 2781	1 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397   ⊆ wss 3899  {cpr 4580  ‘cfv 6490  Vtxcvtx 29018  Edgcedg 29069

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-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rex 3059  df-rab 3398  df-v 3440  df-un 3904  df-ss 3916  df-sn 4579  df-pr 4581

This theorem is referenced by:  dfclnbgr5  48038  dfnbgr5  48039  dfsclnbgr6  48046