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Theorem vopnbgrelself 47727
Description: A vertex 𝑁 is a member of its semiopen neighborhood iff there is a loop joining the vertex with itself. (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
vopnbgrelself (𝑁𝑉 → (𝑁𝑈 ↔ ∃𝑒𝐸 𝑒 = {𝑁}))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑛,𝐸
Allowed substitution hints:   𝑈(𝑒,𝑛)   𝐺(𝑛)

Proof of Theorem vopnbgrelself
StepHypRef Expression
1 ibar 528 . 2 (𝑁𝑉 → (∃𝑒𝐸 ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁})) ↔ (𝑁𝑉 ∧ ∃𝑒𝐸 ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁})))))
2 eqid 2740 . . . . . . 7 𝑁 = 𝑁
32jctl 523 . . . . . 6 (𝑒 = {𝑁} → (𝑁 = 𝑁𝑒 = {𝑁}))
43olcd 873 . . . . 5 (𝑒 = {𝑁} → ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁})))
5 eqneqall 2957 . . . . . . . 8 (𝑁 = 𝑁 → (𝑁𝑁 → ((𝑁𝑒𝑁𝑒) → 𝑒 = {𝑁})))
62, 5ax-mp 5 . . . . . . 7 (𝑁𝑁 → ((𝑁𝑒𝑁𝑒) → 𝑒 = {𝑁}))
763impib 1116 . . . . . 6 ((𝑁𝑁𝑁𝑒𝑁𝑒) → 𝑒 = {𝑁})
8 simpr 484 . . . . . 6 ((𝑁 = 𝑁𝑒 = {𝑁}) → 𝑒 = {𝑁})
97, 8jaoi 856 . . . . 5 (((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁})) → 𝑒 = {𝑁})
104, 9impbii 209 . . . 4 (𝑒 = {𝑁} ↔ ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁})))
1110a1i 11 . . 3 (𝑁𝑉 → (𝑒 = {𝑁} ↔ ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁}))))
1211rexbidv 3185 . 2 (𝑁𝑉 → (∃𝑒𝐸 𝑒 = {𝑁} ↔ ∃𝑒𝐸 ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁}))))
13 dfvopnbgr2.v . . 3 𝑉 = (Vtx‘𝐺)
14 dfvopnbgr2.e . . 3 𝐸 = (Edg‘𝐺)
15 dfvopnbgr2.u . . 3 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
1613, 14, 15vopnbgrel 47726 . 2 (𝑁𝑉 → (𝑁𝑈 ↔ (𝑁𝑉 ∧ ∃𝑒𝐸 ((𝑁𝑁𝑁𝑒𝑁𝑒) ∨ (𝑁 = 𝑁𝑒 = {𝑁})))))
171, 12, 163bitr4rd 312 1 (𝑁𝑉 → (𝑁𝑈 ↔ ∃𝑒𝐸 𝑒 = {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 846  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wrex 3076  {crab 3443  {csn 4648  cfv 6573  (class class class)co 7448  Vtxcvtx 29031  Edgcedg 29082   NeighbVtx cnbgr 29367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-nbgr 29368
This theorem is referenced by: (None)
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