Theorem nbgrval 27120

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 27117	. 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 26789	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6672	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
5	4, 2	syl6reqr 2877	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑉 = (Vtx‘𝑔))
6	5	eleq2d 2900	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
7	6	biimpac 481	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8		fvex 6685	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
9	8	difexi 5234	. . . 4 ⊢ ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10		rabexg 5236	. . . 4 ⊢ (((Vtx‘𝑔) ∖ {𝑘}) ∈ V → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
11	9, 10	mp1i 13	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
12	4, 2	syl6eqr 2876	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
13	12	adantr 483	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14		sneq 4579	. . . . . . 7 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
15	14	adantl 484	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → {𝑘} = {𝑁})
16	13, 15	difeq12d 4102	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
17	16	adantl 484	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18		fveq2 6672	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19		nbgrval.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
20	18, 19	syl6eqr 2876	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
21	20	adantr 483	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → (Edg‘𝑔) = 𝐸)
22	21	adantl 484	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → (Edg‘𝑔) = 𝐸)
23		preq1 4671	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
24	23	sseq1d 4000	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
25	24	adantl 484	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
26	25	adantl 484	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
27	22, 26	rexeqbidv 3404	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	17, 27	rabeqbidv 3487	. . 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 7310	. 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 20	1 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  {crab 3144  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938  {csn 4569  {cpr 4571  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  Vtxcvtx 26783  Edgcedg 26834   NeighbVtx cnbgr 27116

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-nbgr 27117

This theorem is referenced by:  dfnbgr2  27121  dfnbgr3  27122  nbgrel  27124  nbuhgr  27127  nbupgr  27128  nbumgrvtx  27130  nbgr0vtxlem  27139  nbgrnself  27143