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Theorem nbuhgr 28631
Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 28624 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
Hypotheses
Ref Expression
nbuhgr.v 𝑉 = (Vtx‘𝐺)
nbuhgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbuhgr ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺,𝑛   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑒,𝑋,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem nbuhgr
StepHypRef Expression
1 nbuhgr.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbuhgr.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 28624 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
43a1d 25 . 2 (𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
5 df-nel 3048 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
61nbgrnvtx0 28627 . . . . . 6 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
75, 6sylbir 234 . . . . 5 𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
87adantr 482 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9 simpl 484 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → 𝐺 ∈ UHGraph)
109adantr 482 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
112eleq2i 2826 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
1211biimpi 215 . . . . . . . . . . 11 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edguhgr 28420 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
1410, 12, 13syl2an 597 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15 velpw 4608 . . . . . . . . . . . 12 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
161eqcomi 2742 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1716sseq2i 4012 . . . . . . . . . . . 12 (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒𝑉)
1815, 17bitri 275 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒𝑉)
19 sstr 3991 . . . . . . . . . . . . . . 15 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20 prssg 4823 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑋𝑛 ∈ V) → ((𝑁𝑉𝑛𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
2120bicomd 222 . . . . . . . . . . . . . . . . 17 ((𝑁𝑋𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
2221elvd 3482 . . . . . . . . . . . . . . . 16 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
23 simpl 484 . . . . . . . . . . . . . . . 16 ((𝑁𝑉𝑛𝑉) → 𝑁𝑉)
2422, 23syl6bi 253 . . . . . . . . . . . . . . 15 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉𝑁𝑉))
2519, 24syl5com 31 . . . . . . . . . . . . . 14 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → (𝑁𝑋𝑁𝑉))
2625ex 414 . . . . . . . . . . . . 13 ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒𝑉 → (𝑁𝑋𝑁𝑉)))
2726com13 88 . . . . . . . . . . . 12 (𝑁𝑋 → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
2827ad3antlr 730 . . . . . . . . . . 11 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
2918, 28biimtrid 241 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
3014, 29mpd 15 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3130rexlimdva 3156 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3231con3rr3 155 . . . . . . 7 𝑁𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3332expdimp 454 . . . . . 6 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3433ralrimiv 3146 . . . . 5 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35 rabeq0 4385 . . . . 5 ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3634, 35sylibr 233 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
378, 36eqtr4d 2776 . . 3 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3837ex 414 . 2 𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
394, 38pm2.61i 182 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wnel 3047  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cdif 3946  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629  {cpr 4631  cfv 6544  (class class class)co 7409  Vtxcvtx 28287  Edgcedg 28338  UHGraphcuhgr 28347   NeighbVtx cnbgr 28620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-edg 28339  df-uhgr 28349  df-nbgr 28621
This theorem is referenced by:  uhgrnbgr0nb  28642
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