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Theorem dfsclnbgr6 47849
Description: Alternate definition of a semiclosed neighborhood of a vertex as a union of a semiopen neighborhood and the vertex itself if there is a loop at this vertex. (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}))}
dfsclnbgr6.s 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
Assertion
Ref Expression
dfsclnbgr6 (𝑁𝑉𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑛,𝐸   𝑛,𝐺
Allowed substitution hints:   𝑆(𝑒,𝑛)   𝑈(𝑒,𝑛)

Proof of Theorem dfsclnbgr6
StepHypRef Expression
1 simpr 484 . . . . . . . . . . 11 ((𝑁𝑒𝑛𝑒) → 𝑛𝑒)
21anim1i 615 . . . . . . . . . 10 (((𝑁𝑒𝑛𝑒) ∧ 𝑛 = 𝑁) → (𝑛𝑒𝑛 = 𝑁))
32olcd 874 . . . . . . . . 9 (((𝑁𝑒𝑛𝑒) ∧ 𝑛 = 𝑁) → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)))
43expcom 413 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑁𝑒𝑛𝑒) → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁))))
5 3anass 1094 . . . . . . . . . . . 12 ((𝑛𝑁𝑁𝑒𝑛𝑒) ↔ (𝑛𝑁 ∧ (𝑁𝑒𝑛𝑒)))
65biimpri 228 . . . . . . . . . . 11 ((𝑛𝑁 ∧ (𝑁𝑒𝑛𝑒)) → (𝑛𝑁𝑁𝑒𝑛𝑒))
76orcd 873 . . . . . . . . . 10 ((𝑛𝑁 ∧ (𝑁𝑒𝑛𝑒)) → ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})))
87orcd 873 . . . . . . . . 9 ((𝑛𝑁 ∧ (𝑁𝑒𝑛𝑒)) → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)))
98ex 412 . . . . . . . 8 (𝑛𝑁 → ((𝑁𝑒𝑛𝑒) → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁))))
104, 9pm2.61ine 3024 . . . . . . 7 ((𝑁𝑒𝑛𝑒) → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)))
11 3simpc 1150 . . . . . . . . . 10 ((𝑛𝑁𝑁𝑒𝑛𝑒) → (𝑁𝑒𝑛𝑒))
1211a1i 11 . . . . . . . . 9 (𝑁𝑉 → ((𝑛𝑁𝑁𝑒𝑛𝑒) → (𝑁𝑒𝑛𝑒)))
13 vsnid 4662 . . . . . . . . . . . . . . . 16 𝑛 ∈ {𝑛}
1413a1i 11 . . . . . . . . . . . . . . 15 (𝑒 = {𝑛} → 𝑛 ∈ {𝑛})
15 eleq2 2829 . . . . . . . . . . . . . . 15 (𝑒 = {𝑛} → (𝑛𝑒𝑛 ∈ {𝑛}))
1614, 15mpbird 257 . . . . . . . . . . . . . 14 (𝑒 = {𝑛} → 𝑛𝑒)
1716adantl 481 . . . . . . . . . . . . 13 ((𝑛 = 𝑁𝑒 = {𝑛}) → 𝑛𝑒)
18 eleq1 2828 . . . . . . . . . . . . . . 15 (𝑛 = 𝑁 → (𝑛𝑒𝑁𝑒))
1918bicomd 223 . . . . . . . . . . . . . 14 (𝑛 = 𝑁 → (𝑁𝑒𝑛𝑒))
2019adantr 480 . . . . . . . . . . . . 13 ((𝑛 = 𝑁𝑒 = {𝑛}) → (𝑁𝑒𝑛𝑒))
2117, 20mpbird 257 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑒 = {𝑛}) → 𝑁𝑒)
2221adantl 481 . . . . . . . . . . 11 ((𝑁𝑉 ∧ (𝑛 = 𝑁𝑒 = {𝑛})) → 𝑁𝑒)
2317adantl 481 . . . . . . . . . . 11 ((𝑁𝑉 ∧ (𝑛 = 𝑁𝑒 = {𝑛})) → 𝑛𝑒)
2422, 23jca 511 . . . . . . . . . 10 ((𝑁𝑉 ∧ (𝑛 = 𝑁𝑒 = {𝑛})) → (𝑁𝑒𝑛𝑒))
2524ex 412 . . . . . . . . 9 (𝑁𝑉 → ((𝑛 = 𝑁𝑒 = {𝑛}) → (𝑁𝑒𝑛𝑒)))
2612, 25jaod 859 . . . . . . . 8 (𝑁𝑉 → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) → (𝑁𝑒𝑛𝑒)))
2718biimpac 478 . . . . . . . . . 10 ((𝑛𝑒𝑛 = 𝑁) → 𝑁𝑒)
28 simpl 482 . . . . . . . . . 10 ((𝑛𝑒𝑛 = 𝑁) → 𝑛𝑒)
2927, 28jca 511 . . . . . . . . 9 ((𝑛𝑒𝑛 = 𝑁) → (𝑁𝑒𝑛𝑒))
3029a1i 11 . . . . . . . 8 (𝑁𝑉 → ((𝑛𝑒𝑛 = 𝑁) → (𝑁𝑒𝑛𝑒)))
3126, 30jaod 859 . . . . . . 7 (𝑁𝑉 → ((((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)) → (𝑁𝑒𝑛𝑒)))
3210, 31impbid2 226 . . . . . 6 (𝑁𝑉 → ((𝑁𝑒𝑛𝑒) ↔ (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁))))
3332rexbidv 3178 . . . . 5 (𝑁𝑉 → (∃𝑒𝐸 (𝑁𝑒𝑛𝑒) ↔ ∃𝑒𝐸 (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁))))
34 r19.43 3121 . . . . . 6 (∃𝑒𝐸 (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)) ↔ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ ∃𝑒𝐸 (𝑛𝑒𝑛 = 𝑁)))
3534a1i 11 . . . . 5 (𝑁𝑉 → (∃𝑒𝐸 (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)) ↔ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ ∃𝑒𝐸 (𝑛𝑒𝑛 = 𝑁))))
36 r19.41v 3188 . . . . . . . 8 (∃𝑒𝐸 (𝑛𝑒𝑛 = 𝑁) ↔ (∃𝑒𝐸 𝑛𝑒𝑛 = 𝑁))
3736biancomi 462 . . . . . . 7 (∃𝑒𝐸 (𝑛𝑒𝑛 = 𝑁) ↔ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))
3837a1i 11 . . . . . 6 (𝑁𝑉 → (∃𝑒𝐸 (𝑛𝑒𝑛 = 𝑁) ↔ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)))
3938orbi2d 915 . . . . 5 (𝑁𝑉 → ((∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ ∃𝑒𝐸 (𝑛𝑒𝑛 = 𝑁)) ↔ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))))
4033, 35, 393bitrd 305 . . . 4 (𝑁𝑉 → (∃𝑒𝐸 (𝑁𝑒𝑛𝑒) ↔ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))))
4140rabbidv 3443 . . 3 (𝑁𝑉 → {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} = {𝑛𝑉 ∣ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))})
42 unrab 4314 . . . 4 ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)}) = {𝑛𝑉 ∣ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))}
43 rabsneq 4643 . . . . . 6 (𝑁𝑉 → {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒} = {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)})
4443eqcomd 2742 . . . . 5 (𝑁𝑉 → {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)} = {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒})
4544uneq2d 4167 . . . 4 (𝑁𝑉 → ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)}) = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
4642, 45eqtr3id 2790 . . 3 (𝑁𝑉 → {𝑛𝑉 ∣ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))} = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
4741, 46eqtrd 2776 . 2 (𝑁𝑉 → {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
48 dfvopnbgr2.v . . 3 𝑉 = (Vtx‘𝐺)
49 dfsclnbgr6.s . . 3 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
50 dfvopnbgr2.e . . 3 𝐸 = (Edg‘𝐺)
5148, 49, 50dfsclnbgr2 47837 . 2 (𝑁𝑉𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
52 dfvopnbgr2.u . . . 4 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
5348, 50, 52dfvopnbgr2 47844 . . 3 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
5453uneq1d 4166 . 2 (𝑁𝑉 → (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}) = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
5547, 51, 543eqtr4d 2786 1 (𝑁𝑉𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wrex 3069  {crab 3435  cun 3948  wss 3950  {csn 4625  {cpr 4627  cfv 6560  (class class class)co 7432  Vtxcvtx 29014  Edgcedg 29065   NeighbVtx cnbgr 29350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-nbgr 29351
This theorem is referenced by:  dfnbgrss2  47850
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