Theorem nbgrval 29950

Description: The set of neighbors of a vertex 𝑉 in a graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)

Hypotheses

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

Assertion

Ref	Expression
nbgrval	⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃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 nbgrval

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

Step	Hyp	Ref	Expression
1		df-nbgr 29947	. 2 ⊢ NeighbVtx = (𝑔 ∈ V, 𝑘 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒})
2		nbgrval.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3	2	1vgrex 29613	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6911	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	eqtr4id 2817	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑉 = (Vtx‘𝑔))
6	5	eleq2d 2850	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
7	6	biimpac 484	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8		fvex 6924	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
9	8	difexi 5297	. . . 4 ⊢ ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10		rabexg 5304	. . . 4 ⊢ (((Vtx‘𝑔) ∖ {𝑘}) ∈ V → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
11	9, 10	mp1i 14	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
12	4, 2	eqtr4di 2816	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
13	12	adantr 486	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14		sneq 4594	. . . . . . 7 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
15	14	adantl 487	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → {𝑘} = {𝑁})
16	13, 15	difeq12d 4067	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
17	16	adantl 487	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18		fveq2 6911	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19		nbgrval.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
20	18, 19	eqtr4di 2816	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
21	20	adantr 486	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → (Edg‘𝑔) = 𝐸)
22	21	adantl 487	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → (Edg‘𝑔) = 𝐸)
23		preq1 4694	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
24	23	sseq1d 3944	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
25	24	adantl 487	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
26	25	adantl 487	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
27	22, 26	rexeqbidv 3339	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	17, 27	rabeqbidv 3433	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
29	3, 7, 11, 28	ovmpodv2 7605	. 2 ⊢ (𝑁 ∈ 𝑉 → ( NeighbVtx = (𝑔 ∈ V, 𝑘 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒}) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
30	1, 29	mpi 21	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃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   ∖ cdif 4048  {csn 4588  {cpr 4590  ‘cfv 6561  (class class class)co 7446   ∈ cmpo 7448  Vtxcvtx 29607  Edgcedg 29658   NeighbVtx cnbgr 29946

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 5253  ax-nul 5264  ax-pr 5396

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 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-nbgr 29947

This theorem is used by:  dfnbgr2  29951  dfnbgr3  29952  nbgrel  29954  nbuhgr  29957  nbupgr  29958  nbumgrvtx  29960  nbgr0edglem  29970  nbgrnself  29973