Theorem sclnbgrel 49052

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 2852	. 2 ⊢ (𝑋 ∈ 𝑆 ↔ 𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3		preq2 4691	. . . . 5 ⊢ (𝑛 = 𝑋 → {𝑁, 𝑛} = {𝑁, 𝑋})
4	3	sseq1d 3940	. . . 4 ⊢ (𝑛 = 𝑋 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑋} ⊆ 𝑒))
5	4	rexbidv 3186	. . 3 ⊢ (𝑛 = 𝑋 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
6	5	elrab 3644	. 2 ⊢ (𝑋 ∈ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
7	2, 6	bitri 278	1 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  {crab 3412   ⊆ wss 3896  {cpr 4586  ‘cfv 6541  Vtxcvtx 29581  Edgcedg 29632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-un 4150  df-sn 4585  df-pr 4587

This theorem is used by:  sclnbgrelself  49053  sclnbgrisvtx  49054