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Theorem nbuhgr 27124
Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 27117 (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 27117 . . 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 3118 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
61nbgrnvtx0 27120 . . . . . 6 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
75, 6sylbir 238 . . . . 5 𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
87adantr 484 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9 simpl 486 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → 𝐺 ∈ UHGraph)
109adantr 484 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
112eleq2i 2907 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
1211biimpi 219 . . . . . . . . . . 11 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edguhgr 26913 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
1410, 12, 13syl2an 598 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15 velpw 4525 . . . . . . . . . . . 12 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
161eqcomi 2833 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1716sseq2i 3980 . . . . . . . . . . . 12 (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒𝑉)
1815, 17bitri 278 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒𝑉)
19 sstr 3959 . . . . . . . . . . . . . . 15 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20 prssg 4735 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑋𝑛 ∈ V) → ((𝑁𝑉𝑛𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
2120bicomd 226 . . . . . . . . . . . . . . . . 17 ((𝑁𝑋𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
2221elvd 3485 . . . . . . . . . . . . . . . 16 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
23 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑁𝑉𝑛𝑉) → 𝑁𝑉)
2422, 23syl6bi 256 . . . . . . . . . . . . . . 15 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉𝑁𝑉))
2519, 24syl5com 31 . . . . . . . . . . . . . 14 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → (𝑁𝑋𝑁𝑉))
2625ex 416 . . . . . . . . . . . . 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, 28syl5bi 245 . . . . . . . . . 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 3276 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3231con3rr3 158 . . . . . . 7 𝑁𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3332expdimp 456 . . . . . 6 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3433ralrimiv 3175 . . . . 5 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35 rabeq0 4319 . . . . 5 ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3634, 35sylibr 237 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
378, 36eqtr4d 2862 . . 3 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3837ex 416 . 2 𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
394, 38pm2.61i 185 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 209  wa 399   = wceq 1538  wcel 2115  wnel 3117  wral 3132  wrex 3133  {crab 3136  Vcvv 3479  cdif 3915  wss 3918  c0 4274  𝒫 cpw 4520  {csn 4548  {cpr 4550  cfv 6338  (class class class)co 7140  Vtxcvtx 26780  Edgcedg 26831  UHGraphcuhgr 26840   NeighbVtx cnbgr 27113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-nel 3118  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7674  df-2nd 7675  df-edg 26832  df-uhgr 26842  df-nbgr 27114
This theorem is referenced by:  uhgrnbgr0nb  27135
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