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Theorem vopnbgrel 47778
Description: Characterization of a member 𝑋 of the semiopen neighborhood of a vertex 𝑁 in a graph 𝐺. (Contributed by AV, 16-May-2025.)
Hypotheses
Ref Expression
dfvopnbgr2.v 𝑉 = (Vtx‘𝐺)
dfvopnbgr2.e 𝐸 = (Edg‘𝐺)
dfvopnbgr2.u 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
Assertion
Ref Expression
vopnbgrel (𝑁𝑉 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ ∃𝑒𝐸 ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋})))))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑛,𝐸   𝑒,𝑋,𝑛
Allowed substitution hints:   𝑈(𝑒,𝑛)   𝐺(𝑛)

Proof of Theorem vopnbgrel
StepHypRef Expression
1 dfvopnbgr2.v . . . 4 𝑉 = (Vtx‘𝐺)
2 dfvopnbgr2.e . . . 4 𝐸 = (Edg‘𝐺)
3 dfvopnbgr2.u . . . 4 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
41, 2, 3dfvopnbgr2 47777 . . 3 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
54eleq2d 2825 . 2 (𝑁𝑉 → (𝑋𝑈𝑋 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))}))
6 neeq1 3001 . . . . . 6 (𝑛 = 𝑋 → (𝑛𝑁𝑋𝑁))
7 eleq1 2827 . . . . . 6 (𝑛 = 𝑋 → (𝑛𝑒𝑋𝑒))
86, 73anbi13d 1437 . . . . 5 (𝑛 = 𝑋 → ((𝑛𝑁𝑁𝑒𝑛𝑒) ↔ (𝑋𝑁𝑁𝑒𝑋𝑒)))
9 eqeq1 2739 . . . . . 6 (𝑛 = 𝑋 → (𝑛 = 𝑁𝑋 = 𝑁))
10 sneq 4641 . . . . . . 7 (𝑛 = 𝑋 → {𝑛} = {𝑋})
1110eqeq2d 2746 . . . . . 6 (𝑛 = 𝑋 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑋}))
129, 11anbi12d 632 . . . . 5 (𝑛 = 𝑋 → ((𝑛 = 𝑁𝑒 = {𝑛}) ↔ (𝑋 = 𝑁𝑒 = {𝑋})))
138, 12orbi12d 918 . . . 4 (𝑛 = 𝑋 → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ↔ ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋}))))
1413rexbidv 3177 . . 3 (𝑛 = 𝑋 → (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ↔ ∃𝑒𝐸 ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋}))))
1514elrab 3695 . 2 (𝑋 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ↔ (𝑋𝑉 ∧ ∃𝑒𝐸 ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋}))))
165, 15bitrdi 287 1 (𝑁𝑉 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ ∃𝑒𝐸 ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  {csn 4631  cfv 6563  (class class class)co 7431  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  ax-un 7754
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-sbc 3792  df-csb 3909  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-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-nbgr 29365
This theorem is referenced by:  vopnbgrelself  47779
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