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Theorem dfnbgr2 40561
 Description: Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
nbgrval.v 𝑉 = (Vtx‘𝐺)
nbgrval.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
dfnbgr2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑛   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem dfnbgr2
StepHypRef Expression
1 nbgrval.v . . 3 𝑉 = (Vtx‘𝐺)
2 nbgrval.e . . 3 𝐸 = (Edg‘𝐺)
31, 2nbgrval 40560 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4 vex 3176 . . . . . 6 𝑛 ∈ V
5 prssg 4290 . . . . . 6 ((𝑁𝑉𝑛 ∈ V) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
64, 5mpan2 703 . . . . 5 (𝑁𝑉 → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
76bicomd 212 . . . 4 (𝑁𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
87rexbidv 3034 . . 3 (𝑁𝑉 → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)))
98rabbidv 3164 . 2 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
103, 9eqtrd 2644 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   NeighbVtx cnbgr 40550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-nbgr 40554 This theorem is referenced by: (None)
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