Theorem dfclnbgr4 48805

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 2760	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	dfclnbgr2 48804	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}))
4		undif2 4431	. . . 4 ⊢ ({𝑁} ∪ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁})) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
5		rabdif 4267	. . . . 5 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}
6	5	uneq2i 4112	. . . 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 2785	. . 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 29829	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
9	8	eqcomd 2766	. . . 4 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = (𝐺 NeighbVtx 𝑁))
10	9	uneq2d 4115	. . 3 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
11	7, 10	eqtrid 2807	. 2 ⊢ (𝑁 ∈ 𝑉 → ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ (Edg‘𝐺)(𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)}) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
12	3, 11	eqtrd 2795	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412   ∖ cdif 3896   ∪ cun 3897  {csn 4584  ‘cfv 6535  (class class class)co 7416  Vtxcvtx 29485  Edgcedg 29536   NeighbVtx cnbgr 29824   ClNeighbVtx cclnbgr 48799

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6491  df-fun 6537  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-nbgr 29825  df-clnbgr 48800

This theorem is used by:  elclnbgrelnbgr  48806  clnbupgr  48814  clnbgr0edg  48818  edgusgrclnbfin  48823  stgrclnbgr0  48946  isubgr3stgrlem1  48947