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Theorem dfnbgr6 47843
Description: Alternate definition of the (open) neighborhood of a vertex as a difference of its semiopen neighborhood and the singleton of itself. (Contributed by AV, 17-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
dfnbgr6 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑛,𝐸   𝑛,𝐺
Allowed substitution hints:   𝑈(𝑒,𝑛)

Proof of Theorem dfnbgr6
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 rabdif 4321 . . 3 ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)}
2 3anass 1095 . . . . . . . . . . . . 13 ((𝑣𝑁𝑁𝑒𝑣𝑒) ↔ (𝑣𝑁 ∧ (𝑁𝑒𝑣𝑒)))
32biimpri 228 . . . . . . . . . . . 12 ((𝑣𝑁 ∧ (𝑁𝑒𝑣𝑒)) → (𝑣𝑁𝑁𝑒𝑣𝑒))
43orcd 874 . . . . . . . . . . 11 ((𝑣𝑁 ∧ (𝑁𝑒𝑣𝑒)) → ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣})))
54ex 412 . . . . . . . . . 10 (𝑣𝑁 → ((𝑁𝑒𝑣𝑒) → ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))))
6 3simpc 1151 . . . . . . . . . . . 12 ((𝑣𝑁𝑁𝑒𝑣𝑒) → (𝑁𝑒𝑣𝑒))
76a1i 11 . . . . . . . . . . 11 (𝑣𝑁 → ((𝑣𝑁𝑁𝑒𝑣𝑒) → (𝑁𝑒𝑣𝑒)))
8 eqneqall 2951 . . . . . . . . . . . . 13 (𝑣 = 𝑁 → (𝑣𝑁 → (𝑒 = {𝑣} → (𝑁𝑒𝑣𝑒))))
98com12 32 . . . . . . . . . . . 12 (𝑣𝑁 → (𝑣 = 𝑁 → (𝑒 = {𝑣} → (𝑁𝑒𝑣𝑒))))
109impd 410 . . . . . . . . . . 11 (𝑣𝑁 → ((𝑣 = 𝑁𝑒 = {𝑣}) → (𝑁𝑒𝑣𝑒)))
117, 10jaod 860 . . . . . . . . . 10 (𝑣𝑁 → (((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣})) → (𝑁𝑒𝑣𝑒)))
125, 11impbid 212 . . . . . . . . 9 (𝑣𝑁 → ((𝑁𝑒𝑣𝑒) ↔ ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))))
1312rexbidv 3179 . . . . . . . 8 (𝑣𝑁 → (∃𝑒𝐸 (𝑁𝑒𝑣𝑒) ↔ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))))
1413anbi2d 630 . . . . . . 7 (𝑣𝑁 → ((𝑣𝑉 ∧ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)) ↔ (𝑣𝑉 ∧ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣})))))
1514pm5.32ri 575 . . . . . 6 (((𝑣𝑉 ∧ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)) ∧ 𝑣𝑁) ↔ ((𝑣𝑉 ∧ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))) ∧ 𝑣𝑁))
1615a1i 11 . . . . 5 (𝑁𝑉 → (((𝑣𝑉 ∧ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)) ∧ 𝑣𝑁) ↔ ((𝑣𝑉 ∧ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))) ∧ 𝑣𝑁)))
17 eldif 3961 . . . . . 6 (𝑣 ∈ ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}) ↔ (𝑣 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∧ ¬ 𝑣 ∈ {𝑁}))
18 elequ1 2115 . . . . . . . . . 10 (𝑛 = 𝑣 → (𝑛𝑒𝑣𝑒))
1918anbi2d 630 . . . . . . . . 9 (𝑛 = 𝑣 → ((𝑁𝑒𝑛𝑒) ↔ (𝑁𝑒𝑣𝑒)))
2019rexbidv 3179 . . . . . . . 8 (𝑛 = 𝑣 → (∃𝑒𝐸 (𝑁𝑒𝑛𝑒) ↔ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)))
2120elrab 3692 . . . . . . 7 (𝑣 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ↔ (𝑣𝑉 ∧ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)))
22 velsn 4642 . . . . . . . 8 (𝑣 ∈ {𝑁} ↔ 𝑣 = 𝑁)
2322necon3bbii 2988 . . . . . . 7 𝑣 ∈ {𝑁} ↔ 𝑣𝑁)
2421, 23anbi12i 628 . . . . . 6 ((𝑣 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∧ ¬ 𝑣 ∈ {𝑁}) ↔ ((𝑣𝑉 ∧ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)) ∧ 𝑣𝑁))
2517, 24bitri 275 . . . . 5 (𝑣 ∈ ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}) ↔ ((𝑣𝑉 ∧ ∃𝑒𝐸 (𝑁𝑒𝑣𝑒)) ∧ 𝑣𝑁))
26 eldif 3961 . . . . . 6 (𝑣 ∈ ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∖ {𝑁}) ↔ (𝑣 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∧ ¬ 𝑣 ∈ {𝑁}))
27 neeq1 3003 . . . . . . . . . . 11 (𝑛 = 𝑣 → (𝑛𝑁𝑣𝑁))
2827, 183anbi13d 1440 . . . . . . . . . 10 (𝑛 = 𝑣 → ((𝑛𝑁𝑁𝑒𝑛𝑒) ↔ (𝑣𝑁𝑁𝑒𝑣𝑒)))
29 eqeq1 2741 . . . . . . . . . . 11 (𝑛 = 𝑣 → (𝑛 = 𝑁𝑣 = 𝑁))
30 sneq 4636 . . . . . . . . . . . 12 (𝑛 = 𝑣 → {𝑛} = {𝑣})
3130eqeq2d 2748 . . . . . . . . . . 11 (𝑛 = 𝑣 → (𝑒 = {𝑛} ↔ 𝑒 = {𝑣}))
3229, 31anbi12d 632 . . . . . . . . . 10 (𝑛 = 𝑣 → ((𝑛 = 𝑁𝑒 = {𝑛}) ↔ (𝑣 = 𝑁𝑒 = {𝑣})))
3328, 32orbi12d 919 . . . . . . . . 9 (𝑛 = 𝑣 → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ↔ ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))))
3433rexbidv 3179 . . . . . . . 8 (𝑛 = 𝑣 → (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ↔ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))))
3534elrab 3692 . . . . . . 7 (𝑣 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ↔ (𝑣𝑉 ∧ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))))
3635, 23anbi12i 628 . . . . . 6 ((𝑣 ∈ {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∧ ¬ 𝑣 ∈ {𝑁}) ↔ ((𝑣𝑉 ∧ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))) ∧ 𝑣𝑁))
3726, 36bitri 275 . . . . 5 (𝑣 ∈ ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∖ {𝑁}) ↔ ((𝑣𝑉 ∧ ∃𝑒𝐸 ((𝑣𝑁𝑁𝑒𝑣𝑒) ∨ (𝑣 = 𝑁𝑒 = {𝑣}))) ∧ 𝑣𝑁))
3816, 25, 373bitr4g 314 . . . 4 (𝑁𝑉 → (𝑣 ∈ ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}) ↔ 𝑣 ∈ ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∖ {𝑁})))
3938eqrdv 2735 . . 3 (𝑁𝑉 → ({𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} ∖ {𝑁}) = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∖ {𝑁}))
401, 39eqtr3id 2791 . 2 (𝑁𝑉 → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∖ {𝑁}))
41 dfvopnbgr2.v . . 3 𝑉 = (Vtx‘𝐺)
42 dfvopnbgr2.e . . 3 𝐸 = (Edg‘𝐺)
4341, 42dfnbgr2 29354 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
44 dfvopnbgr2.u . . . 4 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
4541, 42, 44dfvopnbgr2 47839 . . 3 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
4645difeq1d 4125 . 2 (𝑁𝑉 → (𝑈 ∖ {𝑁}) = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∖ {𝑁}))
4740, 43, 463eqtr4d 2787 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  cdif 3948  {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:  dfnbgrss2  47845
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