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Theorem nbgrval 26274
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 26270 . 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 25927 . . 3 (𝑁𝑉𝐺 ∈ V)
4 fveq2 6229 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
54, 2syl6reqr 2704 . . . . 5 (𝑔 = 𝐺𝑉 = (Vtx‘𝑔))
65eleq2d 2716 . . . 4 (𝑔 = 𝐺 → (𝑁𝑉𝑁 ∈ (Vtx‘𝑔)))
76biimpac 502 . . 3 ((𝑁𝑉𝑔 = 𝐺) → 𝑁 ∈ (Vtx‘𝑔))
8 fvex 6239 . . . . 5 (Vtx‘𝑔) ∈ V
98difexi 4842 . . . 4 ((Vtx‘𝑔) ∖ {𝑘}) ∈ V
10 rabexg 4844 . . . 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 2703 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1312adantr 480 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → (Vtx‘𝑔) = 𝑉)
14 sneq 4220 . . . . . . 7 (𝑘 = 𝑁 → {𝑘} = {𝑁})
1514adantl 481 . . . . . 6 ((𝑔 = 𝐺𝑘 = 𝑁) → {𝑘} = {𝑁})
1613, 15difeq12d 3762 . . . . 5 ((𝑔 = 𝐺𝑘 = 𝑁) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
1716adantl 481 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → ((Vtx‘𝑔) ∖ {𝑘}) = (𝑉 ∖ {𝑁}))
18 fveq2 6229 . . . . . . . 8 (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺))
19 nbgrval.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
2018, 19syl6eqr 2703 . . . . . . 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 4300 . . . . . . . 8 (𝑘 = 𝑁 → {𝑘, 𝑛} = {𝑁, 𝑛})
2423sseq1d 3665 . . . . . . 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 3183 . . . 4 ((𝑁𝑉 ∧ (𝑔 = 𝐺𝑘 = 𝑁)) → (∃𝑒 ∈ (Edg‘𝑔){𝑘, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
2817, 27rabeqbidv 3226 . . 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 6836 . 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 383   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  {csn 4210  {cpr 4212  cfv 5926  (class class class)co 6690  cmpt2 6692  Vtxcvtx 25919  Edgcedg 25984   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-nbgr 26270
This theorem is referenced by:  dfnbgr2  26275  dfnbgr3  26276  nbgrel  26278  nbgrelOLD  26279  nbuhgr  26284  nbupgr  26285  nbumgrvtx  26287  nbgr0vtxlem  26296  nbgrnself  26300
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