Theorem clnbgrval 47803

Description: The closed neighborhood of a vertex 𝑉 in a graph 𝐺. (Contributed by AV, 7-May-2025.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem clnbgrval

Dummy variables 𝑔 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clnbgr 47800	. 2 ⊢ ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
2		clnbgrval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	1vgrex 28986	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6881	. . . . . . 7 ⊢ (𝐺 = 𝑔 → (Vtx‘𝐺) = (Vtx‘𝑔))
5	4	eqcoms 2744	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝐺) = (Vtx‘𝑔))
6	2, 5	eqtrid 2783	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑉 = (Vtx‘𝑔))
7	6	eleq2d 2821	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
8	7	biimpac 478	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
9		vsnex 5409	. . . . 5 ⊢ {𝑣} ∈ V
10	9	a1i 11	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → {𝑣} ∈ V)
11		fvex 6894	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
12		rabexg 5312	. . . . 5 ⊢ ((Vtx‘𝑔) ∈ V → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} ∈ V)
13	11, 12	mp1i 13	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} ∈ V)
14	10, 13	unexd 7753	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) ∈ V)
15		sneq 4616	. . . . . 6 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
16	15	adantl 481	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → {𝑣} = {𝑁})
17		fveq2 6881	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18	17, 2	eqtr4di 2789	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
19	18	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (Vtx‘𝑔) = 𝑉)
20		fveq2 6881	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
21		clnbgrval.e	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
22	20, 21	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (Edg‘𝑔) = 𝐸)
24		preq1 4714	. . . . . . . . 9 ⊢ (𝑣 = 𝑁 → {𝑣, 𝑛} = {𝑁, 𝑛})
25	24	sseq1d 3995	. . . . . . . 8 ⊢ (𝑣 = 𝑁 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
26	25	adantl 481	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
27	23, 26	rexeqbidv 3330	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	19, 27	rabeqbidv 3439	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
29	16, 28	uneq12d 4149	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
30	29	adantl 481	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
31	3, 8, 14, 30	ovmpodv2 7570	. 2 ⊢ (𝑁 ∈ 𝑉 → ( ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})))
32	1, 31	mpi 20	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  {crab 3420  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931  {csn 4606  {cpr 4608  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Vtxcvtx 28980  Edgcedg 29031   ClNeighbVtx cclnbgr 47799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-clnbgr 47800

This theorem is referenced by:  dfclnbgr2  47804  dfclnbgr3  47807  clnbgrel  47809