Theorem nbgrval 29299

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 29296	. 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 28965	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝐺 ∈ V)
4		fveq2 6826	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
5	2, 4	eqtr4id 2783	. . . . 5 ⊢ (𝑔 = 𝐺 → 𝑉 = (Vtx‘𝑔))
6	5	eleq2d 2814	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝑔)))
7	6	biimpac 478	. . 3 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8		fvex 6839	. . . . 5 ⊢ (Vtx‘𝑔) ∈ V
9	8	difexi 5272	. . . 4 ⊢ ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10		rabexg 5279	. . . 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	eqtr4di 2782	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
13	12	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14		sneq 4589	. . . . . . 7 ⊢ (𝑘 = 𝑁 → {𝑘} = {𝑁})
15	14	adantl 481	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → {𝑘} = {𝑁})
16	13, 15	difeq12d 4080	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
17	16	adantl 481	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18		fveq2 6826	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19		nbgrval.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
20	18, 19	eqtr4di 2782	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
21	20	adantr 480	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → (Edg‘𝑔) = 𝐸)
22	21	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → (Edg‘𝑔) = 𝐸)
23		preq1 4687	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
24	23	sseq1d 3969	. . . . . . 7 ⊢ (𝑘 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
25	24	adantl 481	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑘 = 𝑁) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
26	25	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
27	22, 26	rexeqbidv 3311	. . . 4 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑔 = 𝐺 ∧ 𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
28	17, 27	rabeqbidv 3415	. . 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 7511	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  {crab 3396  Vcvv 3438   ∖ cdif 3902   ⊆ wss 3905  {csn 4579  {cpr 4581  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  Vtxcvtx 28959  Edgcedg 29010   NeighbVtx cnbgr 29295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-nbgr 29296

This theorem is referenced by:  dfnbgr2  29300  dfnbgr3  29301  nbgrel  29303  nbuhgr  29306  nbupgr  29307  nbumgrvtx  29309  nbgr0edglem  29319  nbgrnself  29322