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Theorem dfsclnbgr6 47782
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 3023 . . . . . . 7 ((𝑁𝑒𝑛𝑒) → (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁)))
11 3simpc 1149 . . . . . . . . . 10 ((𝑛𝑁𝑁𝑒𝑛𝑒) → (𝑁𝑒𝑛𝑒))
1211a1i 11 . . . . . . . . 9 (𝑁𝑉 → ((𝑛𝑁𝑁𝑒𝑛𝑒) → (𝑁𝑒𝑛𝑒)))
13 vsnid 4668 . . . . . . . . . . . . . . . 16 𝑛 ∈ {𝑛}
1413a1i 11 . . . . . . . . . . . . . . 15 (𝑒 = {𝑛} → 𝑛 ∈ {𝑛})
15 eleq2 2828 . . . . . . . . . . . . . . 15 (𝑒 = {𝑛} → (𝑛𝑒𝑛 ∈ {𝑛}))
1614, 15mpbird 257 . . . . . . . . . . . . . 14 (𝑒 = {𝑛} → 𝑛𝑒)
1716adantl 481 . . . . . . . . . . . . 13 ((𝑛 = 𝑁𝑒 = {𝑛}) → 𝑛𝑒)
18 eleq1 2827 . . . . . . . . . . . . . . 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 3177 . . . . 5 (𝑁𝑉 → (∃𝑒𝐸 (𝑁𝑒𝑛𝑒) ↔ ∃𝑒𝐸 (((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛𝑒𝑛 = 𝑁))))
34 r19.43 3120 . . . . . 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 3187 . . . . . . . 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 3441 . . 3 (𝑁𝑉 → {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)} = {𝑛𝑉 ∣ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))})
42 unrab 4321 . . . 4 ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)}) = {𝑛𝑉 ∣ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))}
43 rabsneq 4649 . . . . . 6 (𝑁𝑉 → {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒} = {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)})
4443eqcomd 2741 . . . . 5 (𝑁𝑉 → {𝑛𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒)} = {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒})
4544uneq2d 4178 . . . 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 2789 . . 3 (𝑁𝑉 → {𝑛𝑉 ∣ (∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒𝐸 𝑛𝑒))} = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
4741, 46eqtrd 2775 . 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 47770 . 2 (𝑁𝑉𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 (𝑁𝑒𝑛𝑒)})
52 dfvopnbgr2.u . . . 4 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
5348, 50, 52dfvopnbgr2 47777 . . 3 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
5453uneq1d 4177 . 2 (𝑁𝑉 → (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}) = ({𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
5547, 51, 543eqtr4d 2785 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 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  cun 3961  wss 3963  {csn 4631  {cpr 4633  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:  dfnbgrss2  47783
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