Theorem dfclnbgr2 48608

Description: Alternate definition of the closed neighborhood of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 7-May-2025.)

Hypotheses

Ref	Expression
clnbgrval.v	⊢ 𝑉 = (Vtx‘𝐺)
clnbgrval.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
dfclnbgr2	⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))

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

Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem dfclnbgr2

Step	Hyp	Ref	Expression
1		clnbgrval.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		clnbgrval.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	clnbgrval 48607	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
4		prssg 4785	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ V) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
5	4	elvd 3461	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
6	5	bicomd 226	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
7	6	rexbidv 3189	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
8	7	rabbidv 3423	. . 3 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	uneq2d 4122	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
10	3, 9	eqtrd 2798	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  {csn 4589  {cpr 4591  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29346  Edgcedg 29397   ClNeighbVtx cclnbgr 48603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-clnbgr 48604

This theorem is referenced by:  dfclnbgr4  48609  dfclnbgr5  48635  dfclnbgr6  48641