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Theorem nbgrval 29368
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 29365 . 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 29034 . . 3 (𝑁𝑉𝐺 ∈ V)
4 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
52, 4eqtr4id 2794 . . . . 5 (𝑔 = 𝐺𝑉 = (Vtx‘𝑔))
65eleq2d 2825 . . . 4 (𝑔 = 𝐺 → (𝑁𝑉𝑁 ∈ (Vtx‘𝑔)))
76biimpac 478 . . 3 ((𝑁𝑉𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8 fvex 6920 . . . . 5 (Vtx‘𝑔) ∈ V
98difexi 5336 . . . 4 ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10 rabexg 5343 . . . 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 2793 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1312adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14 sneq 4641 . . . . . . 7 (𝑘 = 𝑁 → {𝑘} = {𝑁})
1514adantl 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → {𝑘} = {𝑁})
1613, 15difeq12d 4137 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
1716adantl 481 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18 fveq2 6907 . . . . . . . 8 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19 nbgrval.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
2018, 19eqtr4di 2793 . . . . . . 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 4738 . . . . . . . 8 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2423sseq1d 4027 . . . . . . 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 3345 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
2817, 27rabeqbidv 3452 . . 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 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  {csn 4631  {cpr 4633  cfv 6563  (class class class)co 7431  cmpo 7433  Vtxcvtx 29028  Edgcedg 29079   NeighbVtx cnbgr 29364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-nbgr 29365
This theorem is referenced by:  dfnbgr2  29369  dfnbgr3  29370  nbgrel  29372  nbuhgr  29375  nbupgr  29376  nbumgrvtx  29378  nbgr0edglem  29388  nbgrnself  29391
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