Theorem sclnbgrel 48345

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 2832	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3		preq2 4673	. . . . 5 ⊢ (𝑛 = 𝑋 → {𝑁, 𝑛} = {𝑁, 𝑋})
4	3	sseq1d 3953	. . . 4 ⊢ (𝑛 = 𝑋 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑋} ⊆ 𝑒))
5	4	rexbidv 3164	. . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
6	5	elrab 3636	. 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
7	2, 6	bitri 276	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  {crab 3392   ⊆ wss 3890  {cpr 4564  ‘cfv 6492  Vtxcvtx 29090  Edgcedg 29141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065  df-rab 3393  df-v 3434  df-un 3895  df-ss 3907  df-sn 4563  df-pr 4565

This theorem is referenced by:  sclnbgrelself  48346  sclnbgrisvtx  48347