Theorem sclnbgrel 47851

Description: Characterization of a member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 in a graph 𝐺. (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
sclnbgrel	⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))

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

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

Proof of Theorem sclnbgrel

Step	Hyp	Ref	Expression
1		dfsclnbgr2.s	. . 3 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
2	1	eleq2i 2821	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3		preq2 4701	. . . . 5 ⊢ (𝑛 = 𝑋 → {𝑁, 𝑛} = {𝑁, 𝑋})
4	3	sseq1d 3981	. . . 4 ⊢ (𝑛 = 𝑋 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑋} ⊆ 𝑒))
5	4	rexbidv 3158	. . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
6	5	elrab 3662	. 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
7	2, 6	bitri 275	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  {crab 3408   ⊆ wss 3917  {cpr 4594  ‘cfv 6514  Vtxcvtx 28930  Edgcedg 28981

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-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-rab 3409  df-v 3452  df-un 3922  df-ss 3934  df-sn 4593  df-pr 4595

This theorem is referenced by:  sclnbgrelself  47852  sclnbgrisvtx  47853