Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nbuhgr Structured version   Visualization version   GIF version

Theorem nbuhgr 40565
 Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 40560 (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
nbgrel.v 𝑉 = (Vtx‘𝐺)
nbgrel.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,𝑛   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉   𝑛,𝐺   𝑛,𝑁   𝑛,𝑉   𝑒,𝑋,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem nbuhgr
StepHypRef Expression
1 nbgrel.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbgrel.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 40560 . . 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 2783 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
61nbgrnvtx0 40563 . . . . . 6 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
75, 6sylbir 224 . . . . 5 𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
87adantr 480 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9 simpl 472 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → 𝐺 ∈ UHGraph )
109adantr 480 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph )
112eleq2i 2680 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
1211biimpi 205 . . . . . . . . . . 11 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edguhgr 25803 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
1410, 12, 13syl2an 493 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15 selpw 4115 . . . . . . . . . . . 12 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
161eqcomi 2619 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1716sseq2i 3593 . . . . . . . . . . . 12 (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒𝑉)
1815, 17bitri 263 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒𝑉)
19 sstr 3576 . . . . . . . . . . . . . . 15 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20 vex 3176 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
21 prssg 4290 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑋𝑛 ∈ V) → ((𝑁𝑉𝑛𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
2221bicomd 212 . . . . . . . . . . . . . . . . 17 ((𝑁𝑋𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
2320, 22mpan2 703 . . . . . . . . . . . . . . . 16 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
24 simpl 472 . . . . . . . . . . . . . . . 16 ((𝑁𝑉𝑛𝑉) → 𝑁𝑉)
2523, 24syl6bi 242 . . . . . . . . . . . . . . 15 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉𝑁𝑉))
2619, 25syl5com 31 . . . . . . . . . . . . . 14 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → (𝑁𝑋𝑁𝑉))
2726ex 449 . . . . . . . . . . . . 13 ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒𝑉 → (𝑁𝑋𝑁𝑉)))
2827com13 86 . . . . . . . . . . . 12 (𝑁𝑋 → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
2928ad3antlr 763 . . . . . . . . . . 11 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
3018, 29syl5bi 231 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
3114, 30mpd 15 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3231rexlimdva 3013 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3332con3rr3 150 . . . . . . 7 𝑁𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3433expdimp 452 . . . . . 6 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3534ralrimiv 2948 . . . . 5 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
36 rabeq0 3911 . . . . 5 ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3735, 36sylibr 223 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
388, 37eqtr4d 2647 . . 3 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3938ex 449 . 2 𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
404, 39pm2.61i 175 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   UHGraph cuhgr 25722  Edgcedga 25792   NeighbVtx cnbgr 40550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-uhgr 25724  df-edga 25793  df-nbgr 40554 This theorem is referenced by:  uhgrnbgr0nb  40576
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