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Theorem nbgrval 29353
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 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {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.
StepHypRef Expression
1 df-nbgr 29350 . 2 NeighbVtx = (𝑔 ∈ V, 𝑘 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒})
2 nbgrval.v . . . 4 𝑉 = (Vtx‘𝐺)
321vgrex 29019 . . 3 (𝑁𝑉𝐺 ∈ V)
4 fveq2 6906 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
52, 4eqtr4id 2796 . . . . 5 (𝑔 = 𝐺𝑉 = (Vtx‘𝑔))
65eleq2d 2827 . . . 4 (𝑔 = 𝐺 → (𝑁𝑉𝑁 ∈ (Vtx‘𝑔)))
76biimpac 478 . . 3 ((𝑁𝑉𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8 fvex 6919 . . . . 5 (Vtx‘𝑔) ∈ V
98difexi 5330 . . . 4 ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10 rabexg 5337 . . . 4 (((Vtx‘𝑔) ∖ {𝑘}) ∈ V → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
119, 10mp1i 13 . . 3 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} ∈ V)
124, 2eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1312adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14 sneq 4636 . . . . . . 7 (𝑘 = 𝑁 → {𝑘} = {𝑁})
1514adantl 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → {𝑘} = {𝑁})
1613, 15difeq12d 4127 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
1716adantl 481 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18 fveq2 6906 . . . . . . . 8 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19 nbgrval.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
2018, 19eqtr4di 2795 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
2120adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Edg‘𝑔) = 𝐸)
2221adantl 481 . . . . 5 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (Edg‘𝑔) = 𝐸)
23 preq1 4733 . . . . . . . 8 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2423sseq1d 4015 . . . . . . 7 (𝑘 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2524adantl 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2625adantl 481 . . . . 5 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2722, 26rexeqbidv 3347 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
2817, 27rabeqbidv 3455 . . 3 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
293, 7, 11, 28ovmpodv2 7591 . 2 (𝑁𝑉 → ( NeighbVtx = (𝑔 ∈ V, 𝑘 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑘}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒}) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
301, 29mpi 20 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  {csn 4626  {cpr 4628  cfv 6561  (class class class)co 7431  cmpo 7433  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-nbgr 29350
This theorem is referenced by:  dfnbgr2  29354  dfnbgr3  29355  nbgrel  29357  nbuhgr  29360  nbupgr  29361  nbumgrvtx  29363  nbgr0edglem  29373  nbgrnself  29376
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