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Theorem dfsclnbgr2 47770
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 47772), 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 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
dfsclnbgr2.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
dfsclnbgr2 (𝑁𝑉𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
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
StepHypRef Expression
1 dfsclnbgr2.s . 2 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
2 prssg 4824 . . . . 5 ((𝑁𝑉𝑛𝑉) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
32bicomd 223 . . . 4 ((𝑁𝑉𝑛𝑉) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
43rexbidv 3177 . . 3 ((𝑁𝑉𝑛𝑉) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)))
54rabbidva 3440 . 2 (𝑁𝑉 → {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
61, 5eqtrid 2787 1 (𝑁𝑉𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  wss 3963  {cpr 4633  cfv 6563  Vtxcvtx 29028  Edgcedg 29079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634
This theorem is referenced by:  dfclnbgr5  47774  dfnbgr5  47775  dfsclnbgr6  47782
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