Theorem dfclnbgr2 48476

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 48475	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
4		prssg 4789	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ V) → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
5	4	elvd 3469	. . . . . 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 3195	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
8	7	rabbidv 3430	. . 3 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	uneq2d 4130	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
10	3, 9	eqtrd 2804	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  {csn 4594  {cpr 4596  ‘cfv 6537  (class class class)co 7411  Vtxcvtx 29286  Edgcedg 29337   ClNeighbVtx cclnbgr 48471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-clnbgr 48472

This theorem is referenced by:  dfclnbgr4  48477  dfclnbgr5  48503  dfclnbgr6  48509