Theorem clnbgrval 49125

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 49122	. 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 29610	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6908	. . . . . . 7 ⊢ (𝐺 = 𝑔 → (Vtx‘𝐺) = (Vtx‘𝑔))
5	4	eqcoms 2771	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝐺) = (Vtx‘𝑔))
6	2, 5	eqtrid 2810	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑉 = (Vtx‘𝑔))
7	6	eleq2d 2850	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
8	7	biimpac 484	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
9		vsnex 5397	. . . . 5 ⊢ {𝑣} ∈ V
10	9	a1i 11	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → {𝑣} ∈ V)
11		fvex 6921	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
12		rabexg 5303	. . . . 5 ⊢ ((Vtx‘𝑔) ∈ V → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} ∈ V)
13	11, 12	mp1i 14	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒} ∈ V)
14	10, 13	unexd 7792	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑣 = 𝑁)) → ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}) ∈ V)
15		sneq 4594	. . . . . 6 ⊢ (𝑣 = 𝑁 → {𝑣} = {𝑁})
16	15	adantl 487	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → {𝑣} = {𝑁})
17		fveq2 6908	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
18	17, 2	eqtr4di 2816	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
19	18	adantr 486	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (Vtx‘𝑔) = 𝑉)
20		fveq2 6908	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
21		clnbgrval.e	. . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺)
22	20, 21	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
23	22	adantr 486	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (Edg‘𝑔) = 𝐸)
24		preq1 4694	. . . . . . . . 9 ⊢ (𝑣 = 𝑁 → {𝑣, 𝑛} = {𝑁, 𝑛})
25	24	sseq1d 3944	. . . . . . . 8 ⊢ (𝑣 = 𝑁 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
26	25	adantl 487	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
27	23, 26	rexeqbidv 3339	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑁) → (∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	19, 27	rabeqbidv 3433	. . . . 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 4171	. . . 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 487	. . 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 7602	. 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 21	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  {csn 4588  {cpr 4590  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  Vtxcvtx 29604  Edgcedg 29655   ClNeighbVtx cclnbgr 49121

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-clnbgr 49122

This theorem is used by:  dfclnbgr2  49126  dfclnbgr3  49129  clnbgrel  49131