Theorem clnbgrval 48064

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 48061	. 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 29075	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6834	. . . . . . 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 2822	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
8	7	biimpac 478	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
9		vsnex 5379	. . . . 5 ⊢ {𝑣} ∈ V
10	9	a1i 11	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → {𝑣} ∈ V)
11		fvex 6847	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
12		rabexg 5282	. . . . 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 7699	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) ∈ V)
15		sneq 4590	. . . . . 6 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
16	15	adantl 481	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → {𝑣} = {𝑁})
17		fveq2 6834	. . . . . . . 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 6834	. . . . . . . . 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 4690	. . . . . . . . 9 ⊢ (𝑣 = 𝑁 → {𝑣, 𝑛} = {𝑁, 𝑛})
25	24	sseq1d 3965	. . . . . . . 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 3317	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	19, 27	rabeqbidv 3417	. . . . 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 4121	. . . 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 7516	. 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 1541   ∈ wcel 2113  ∃wrex 3060  {crab 3399  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  {csn 4580  {cpr 4582  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Vtxcvtx 29069  Edgcedg 29120   ClNeighbVtx cclnbgr 48060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-clnbgr 48061

This theorem is referenced by:  dfclnbgr2  48065  dfclnbgr3  48068  clnbgrel  48070