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Theorem dfvopnbgr2 47777
Description: Alternate definition of the semiopen neighborhood of a vertex breaking up the subset relationship of an unordered pair. A semiopen neighborhood 𝑈 of a vertex 𝑁 is its open neighborhood together with itself if there is a loop at this vertex. (Contributed by AV, 15-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
dfvopnbgr2 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛
Allowed substitution hints:   𝑈(𝑒,𝑛)   𝐸(𝑛)   𝐺(𝑛)

Proof of Theorem dfvopnbgr2
StepHypRef Expression
1 dfvopnbgr2.u . 2 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
2 dfvopnbgr2.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
3 dfvopnbgr2.e . . . . . . . 8 𝐸 = (Edg‘𝐺)
42, 3nbgrel 29372 . . . . . . 7 (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
54a1i 11 . . . . . 6 ((𝑁𝑉𝑛𝑉) → (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ↔ ((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)))
65orbi1d 916 . . . . 5 ((𝑁𝑉𝑛𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))))
7 df-3an 1088 . . . . . . . . 9 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ↔ (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
8 r19.42v 3189 . . . . . . . . 9 (∃𝑒𝐸 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
97, 8bitr4i 278 . . . . . . . 8 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ↔ ∃𝑒𝐸 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒))
109orbi1i 913 . . . . . . 7 ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ (∃𝑒𝐸 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})))
1110a1i 11 . . . . . 6 ((𝑁𝑉𝑛𝑉) → ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ (∃𝑒𝐸 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))))
12 r19.43 3120 . . . . . 6 (∃𝑒𝐸 ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛𝑒 = {𝑁})) ↔ (∃𝑒𝐸 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})))
1311, 12bitr4di 289 . . . . 5 ((𝑁𝑉𝑛𝑉) → ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ ∃𝑒𝐸 ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛𝑒 = {𝑁}))))
146, 13bitrd 279 . . . 4 ((𝑁𝑉𝑛𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ ∃𝑒𝐸 ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛𝑒 = {𝑁}))))
15 anass 468 . . . . . . . 8 ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛𝑉𝑁𝑉) ∧ (𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒)))
1615a1i 11 . . . . . . 7 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛𝑉𝑁𝑉) ∧ (𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
17 ibar 528 . . . . . . . . 9 ((𝑛𝑉𝑁𝑉) → ((𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛𝑉𝑁𝑉) ∧ (𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
1817ancoms 458 . . . . . . . 8 ((𝑁𝑉𝑛𝑉) → ((𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛𝑉𝑁𝑉) ∧ (𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
1918adantr 480 . . . . . . 7 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ((𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ ((𝑛𝑉𝑁𝑉) ∧ (𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒))))
20 prssg 4824 . . . . . . . . . . 11 ((𝑁𝑉𝑛𝑉) → ((𝑁𝑒𝑛𝑒) ↔ {𝑁, 𝑛} ⊆ 𝑒))
2120bicomd 223 . . . . . . . . . 10 ((𝑁𝑉𝑛𝑉) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
2221adantr 480 . . . . . . . . 9 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ (𝑁𝑒𝑛𝑒)))
2322anbi2d 630 . . . . . . . 8 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ((𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛𝑁 ∧ (𝑁𝑒𝑛𝑒))))
24 3anass 1094 . . . . . . . 8 ((𝑛𝑁𝑁𝑒𝑛𝑒) ↔ (𝑛𝑁 ∧ (𝑁𝑒𝑛𝑒)))
2523, 24bitr4di 289 . . . . . . 7 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ((𝑛𝑁 ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛𝑁𝑁𝑒𝑛𝑒)))
2616, 19, 253bitr2d 307 . . . . . 6 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ↔ (𝑛𝑁𝑁𝑒𝑛𝑒)))
27 eqcom 2742 . . . . . . . . 9 (𝑁 = 𝑛𝑛 = 𝑁)
2827anbi1i 624 . . . . . . . 8 ((𝑁 = 𝑛𝑒 = {𝑁}) ↔ (𝑛 = 𝑁𝑒 = {𝑁}))
29 sneq 4641 . . . . . . . . . . 11 (𝑁 = 𝑛 → {𝑁} = {𝑛})
3029eqcoms 2743 . . . . . . . . . 10 (𝑛 = 𝑁 → {𝑁} = {𝑛})
3130eqeq2d 2746 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑒 = {𝑁} ↔ 𝑒 = {𝑛}))
3231pm5.32i 574 . . . . . . . 8 ((𝑛 = 𝑁𝑒 = {𝑁}) ↔ (𝑛 = 𝑁𝑒 = {𝑛}))
3328, 32bitri 275 . . . . . . 7 ((𝑁 = 𝑛𝑒 = {𝑁}) ↔ (𝑛 = 𝑁𝑒 = {𝑛}))
3433a1i 11 . . . . . 6 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → ((𝑁 = 𝑛𝑒 = {𝑁}) ↔ (𝑛 = 𝑁𝑒 = {𝑛})))
3526, 34orbi12d 918 . . . . 5 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → (((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛𝑒 = {𝑁})) ↔ ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))))
3635rexbidva 3175 . . . 4 ((𝑁𝑉𝑛𝑉) → (∃𝑒𝐸 ((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛𝑒 = {𝑁})) ↔ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))))
3714, 36bitrd 279 . . 3 ((𝑁𝑉𝑛𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))))
3837rabbidva 3440 . 2 (𝑁𝑉 → {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))} = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
391, 38eqtrid 2787 1 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
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  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:  vopnbgrel  47778  dfclnbgr6  47780  dfnbgr6  47781  dfsclnbgr6  47782
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