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Theorem nbgrval 26115
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 26109 . 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 25777 . . 3 (𝑁𝑉𝐺 ∈ V)
4 fveq2 6150 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
54, 2syl6reqr 2679 . . . . 5 (𝑔 = 𝐺𝑉 = (Vtx‘𝑔))
65eleq2d 2689 . . . 4 (𝑔 = 𝐺 → (𝑁𝑉𝑁 ∈ (Vtx‘𝑔)))
76biimpac 503 . . 3 ((𝑁𝑉𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8 fvex 6160 . . . . 5 (Vtx‘𝑔) ∈ V
98difexi 4774 . . . 4 ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10 rabexg 4777 . . . 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, 2syl6eqr 2678 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1312adantr 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14 sneq 4163 . . . . . . 7 (𝑘 = 𝑁 → {𝑘} = {𝑁})
1514adantl 482 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → {𝑘} = {𝑁})
1613, 15difeq12d 3712 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
1716adantl 482 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18 fveq2 6150 . . . . . . . 8 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19 nbgrval.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
2018, 19syl6eqr 2678 . . . . . . 7 (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸)
2120adantr 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Edg‘𝑔) = 𝐸)
2221adantl 482 . . . . 5 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (Edg‘𝑔) = 𝐸)
23 preq1 4243 . . . . . . . 8 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2423sseq1d 3616 . . . . . . 7 (𝑘 = 𝑁 → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2524adantl 482 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2625adantl 482 . . . . 5 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ({𝑘, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ 𝑒))
2722, 26rexeqbidv 3147 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
2817, 27rabeqbidv 3186 . . 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, 28ovmpt2dv2 6748 . 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 196  wa 384   = wceq 1480  wcel 1992  wrex 2913  {crab 2916  Vcvv 3191  cdif 3557  wss 3560  {csn 4153  {cpr 4155  cfv 5850  (class class class)co 6605  cmpt2 6607  Vtxcvtx 25769  Edgcedg 25834   NeighbVtx cnbgr 26105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-nbgr 26109
This theorem is referenced by:  dfnbgr2  26116  dfnbgr3  26117  nbgrel  26119  nbuhgr  26120  nbupgr  26121  nbumgrvtx  26123  nbgr0vtxlem  26132  nbgrnself  26138
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