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Theorem dfvopnbgr2 47839
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 29357 . . . . . . 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 917 . . . . 5 ((𝑁𝑉𝑛𝑉) → ((𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁})) ↔ (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))))
7 df-3an 1089 . . . . . . . . 9 (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁 ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒) ↔ (((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
8 r19.42v 3191 . . . . . . . . 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 914 . . . . . . 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 3122 . . . . . 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 4819 . . . . . . . . . . 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 1095 . . . . . . . 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 2744 . . . . . . . . 9 (𝑁 = 𝑛𝑛 = 𝑁)
2827anbi1i 624 . . . . . . . 8 ((𝑁 = 𝑛𝑒 = {𝑁}) ↔ (𝑛 = 𝑁𝑒 = {𝑁}))
29 sneq 4636 . . . . . . . . . . 11 (𝑁 = 𝑛 → {𝑁} = {𝑛})
3029eqcoms 2745 . . . . . . . . . 10 (𝑛 = 𝑁 → {𝑁} = {𝑛})
3130eqeq2d 2748 . . . . . . . . 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 919 . . . . 5 (((𝑁𝑉𝑛𝑉) ∧ 𝑒𝐸) → (((((𝑛𝑉𝑁𝑉) ∧ 𝑛𝑁) ∧ {𝑁, 𝑛} ⊆ 𝑒) ∨ (𝑁 = 𝑛𝑒 = {𝑁})) ↔ ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))))
3635rexbidva 3177 . . . 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 3443 . 2 (𝑁𝑉 → {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))} = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
391, 38eqtrid 2789 1 (𝑁𝑉𝑈 = {𝑛𝑉 ∣ ∃𝑒𝐸 ((𝑛𝑁𝑁𝑒𝑛𝑒) ∨ (𝑛 = 𝑁𝑒 = {𝑛}))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  wss 3951  {csn 4626  {cpr 4628  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:  vopnbgrel  47840  dfclnbgr6  47842  dfnbgr6  47843  dfsclnbgr6  47844
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