Theorem nbuhgr 29630

Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 29623 (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 ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {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

Step	Hyp	Ref	Expression
1		nbuhgr.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		nbuhgr.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	nbgrval 29623	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4	3	a1d 26	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
5		df-nel 3071	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 ↔ ¬ 𝑁 ∈ 𝑉)
6	1	nbgrnvtx0 29626	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
7	5, 6	sylbir 238	. . . . 5 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
8	7	adantr 485	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → 𝐺 ∈ UHGraph)
10	9	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
11	2	eleq2i 2861	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
13		edguhgr 29416	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
14	10, 12, 13	syl2an 607	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15		velpw 4569	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
16	1	eqcomi 2778	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = 𝑉
17	16	sseq2i 3974	. . . . . . . . . . . 12 ⊢ (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
18	15, 17	bitri 278	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
19		sstr 3953	. . . . . . . . . . . . . . 15 ⊢ (({𝑁, 𝑛} ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20		prssg 4786	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
21	20	bicomd 226	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
22	21	elvd 3469	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
23		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → 𝑁 ∈ 𝑉)
24	22, 23	biimtrdi 256	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 → 𝑁 ∈ 𝑉))
25	19, 24	syl5com 32	. . . . . . . . . . . . . 14 ⊢ (({𝑁, 𝑛} ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉) → (𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉))
26	25	ex 417	. . . . . . . . . . . . 13 ⊢ ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒 ⊆ 𝑉 → (𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉)))
27	26	com13 89	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑋 → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
28	27	ad3antlr 743	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
29	18, 28	biimtrid 245	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
30	14, 29	mpd 16	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
31	30	rexlimdva 3172	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
32	31	con3rr3 156	. . . . . . 7 ⊢ (¬ 𝑁 ∈ 𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
33	32	expdimp 457	. . . . . 6 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
34	33	ralrimiv 3162	. . . . 5 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35		rabeq0 4351	. . . . 5 ⊢ ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
36	34, 35	sylibr 237	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
37	8, 36	eqtr4d 2807	. . 3 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
38	37	ex 417	. 2 ⊢ (¬ 𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
39	4, 38	pm2.61i 184	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∉ wnel 3070  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4564  {csn 4591  {cpr 4593  ‘cfv 6533  (class class class)co 7408  Vtxcvtx 29283  Edgcedg 29334  UHGraphcuhgr 29343   NeighbVtx cnbgr 29619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-edg 29335  df-uhgr 29345  df-nbgr 29620

This theorem is referenced by:  uhgrnbgr0nb  29641