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Theorem vopnbgrel 47840
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 47839 . . 3 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
54eleq2d 2827 . 2 (𝑁𝑉 → (𝑋𝑈𝑋 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))}))
6 neeq1 3003 . . . . . 6 (𝑛 = 𝑋 → (𝑛𝑁𝑋𝑁))
7 eleq1 2829 . . . . . 6 (𝑛 = 𝑋 → (𝑛𝑒𝑋𝑒))
86, 73anbi13d 1440 . . . . 5 (𝑛 = 𝑋 → ((𝑛𝑁𝑁𝑒𝑛𝑒) ↔ (𝑋𝑁𝑁𝑒𝑋𝑒)))
9 eqeq1 2741 . . . . . 6 (𝑛 = 𝑋 → (𝑛 = 𝑁𝑋 = 𝑁))
10 sneq 4636 . . . . . . 7 (𝑛 = 𝑋 → {𝑛} = {𝑋})
1110eqeq2d 2748 . . . . . 6 (𝑛 = 𝑋 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑋}))
129, 11anbi12d 632 . . . . 5 (𝑛 = 𝑋 → ((𝑛 = 𝑁𝑒 = {𝑛}) ↔ (𝑋 = 𝑁𝑒 = {𝑋})))
138, 12orbi12d 919 . . . 4 (𝑛 = 𝑋 → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ↔ ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋}))))
1413rexbidv 3179 . . 3 (𝑛 = 𝑋 → (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ↔ ∃𝑒𝐸 ((𝑋𝑁𝑁𝑒𝑋𝑒) ∨ (𝑋 = 𝑁𝑒 = {𝑋}))))
1514elrab 3692 . 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  {csn 4626  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-nbgr 29350
This theorem is referenced by:  vopnbgrelself  47841
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