Theorem clnbgrval 47747

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 47744	. 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 29034	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6907	. . . . . . 7 ⊢ (𝐺 = 𝑔 → (Vtx‘𝐺) = (Vtx‘𝑔))
5	4	eqcoms 2743	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝐺) = (Vtx‘𝑔))
6	2, 5	eqtrid 2787	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑉 = (Vtx‘𝑔))
7	6	eleq2d 2825	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
8	7	biimpac 478	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
9		vsnex 5440	. . . . 5 ⊢ {𝑣} ∈ V
10	9	a1i 11	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → {𝑣} ∈ V)
11		fvex 6920	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
12		rabexg 5343	. . . . 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 7773	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) ∈ V)
15		sneq 4641	. . . . . 6 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
16	15	adantl 481	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → {𝑣} = {𝑁})
17		fveq2 6907	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18	17, 2	eqtr4di 2793	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
19	18	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (Vtx‘𝑔) = 𝑉)
20		fveq2 6907	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
21		clnbgrval.e	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
22	20, 21	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (Edg‘𝑔) = 𝐸)
24		preq1 4738	. . . . . . . . 9 ⊢ (𝑣 = 𝑁 → {𝑣, 𝑛} = {𝑁, 𝑛})
25	24	sseq1d 4027	. . . . . . . 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 3345	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	19, 27	rabeqbidv 3452	. . . . 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 4179	. . . 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 7591	. 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 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  {csn 4631  {cpr 4633  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Vtxcvtx 29028  Edgcedg 29079   ClNeighbVtx cclnbgr 47743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-clnbgr 47744

This theorem is referenced by:  dfclnbgr2  47748  dfclnbgr3  47751  clnbgrel  47753