Theorem dfclnbgr4 48617

Description: Alternate definition of the closed neighborhood of a vertex as union of the vertex with its open neighborhood. (Contributed by AV, 8-May-2025.)

Hypothesis

Ref	Expression
dfclnbgr4.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
dfclnbgr4	⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

Proof of Theorem dfclnbgr4

Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfclnbgr4.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2763	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	dfclnbgr2 48616	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
4		undif2 4438	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
5		rabdif 4274	. . . . 5 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
6	5	uneq2i 4119	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
7	4, 6	eqtr3i 2788	. . 3 ⊢ ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
8	1, 2	dfnbgr2 29696	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	eqcomd 2769	. . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = (𝐺 NeighbVtx 𝑁))
10	9	uneq2d 4122	. . 3 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
11	7, 10	eqtrid 2810	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
12	3, 11	eqtrd 2798	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416   ∖ cdif 3902   ∪ cun 3903  {csn 4589  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29355  Edgcedg 29406   NeighbVtx cnbgr 29691   ClNeighbVtx cclnbgr 48611

This proof depends on 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 proof 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-nbgr 29692  df-clnbgr 48612

This theorem is used by:  elclnbgrelnbgr  48618  clnbupgr  48626  clnbgr0edg  48630  edgusgrclnbfin  48635  stgrclnbgr0  48758  isubgr3stgrlem1  48759