Theorem nbuhgr 29277

Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 29270 (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 29270	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
4	3	a1d 25	. 2 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
5		df-nel 3031	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 ↔ ¬ 𝑁 ∈ 𝑉)
6	1	nbgrnvtx0 29273	. . . . . 6 ⊢ (𝑁 ∉ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
7	5, 6	sylbir 235	. . . . 5 ⊢ (¬ 𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
8	7	adantr 480	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → 𝐺 ∈ UHGraph)
10	9	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph)
11	2	eleq2i 2821	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝐸 ↔ 𝑒 ∈ (Edg‘𝐺))
12	11	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝐸 → 𝑒 ∈ (Edg‘𝐺))
13		edguhgr 29063	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
14	10, 12, 13	syl2an 596	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15		velpw 4571	. . . . . . . . . . . 12 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
16	1	eqcomi 2739	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = 𝑉
17	16	sseq2i 3979	. . . . . . . . . . . 12 ⊢ (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
18	15, 17	bitri 275	. . . . . . . . . . 11 ⊢ (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ 𝑉)
19		sstr 3958	. . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
22	21	elvd 3456	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉)))
23		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉) → 𝑁 ∈ 𝑉)
24	22, 23	biimtrdi 253	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ 𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 → 𝑁 ∈ 𝑉))
25	19, 24	syl5com 31	. . . . . . . . . . . . . 14 ⊢ (({𝑁, 𝑛} ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉) → (𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉))
26	25	ex 412	. . . . . . . . . . . . 13 ⊢ ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒 ⊆ 𝑉 → (𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉)))
27	26	com13 88	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑋 → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
28	27	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ⊆ 𝑉 → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
29	18, 28	biimtrid 242	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉)))
30	14, 29	mpd 15	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒 ∈ 𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
31	30	rexlimdva 3135	. . . . . . . 8 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒 → 𝑁 ∈ 𝑉))
32	31	con3rr3 155	. . . . . . 7 ⊢ (¬ 𝑁 ∈ 𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
33	32	expdimp 452	. . . . . 6 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒))
34	33	ralrimiv 3125	. . . . 5 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
35		rabeq0 4354	. . . . 5 ⊢ ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒)
36	34, 35	sylibr 234	. . . 4 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
37	8, 36	eqtr4d 2768	. . 3 ⊢ ((¬ 𝑁 ∈ 𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})
38	37	ex 412	. 2 ⊢ (¬ 𝑁 ∈ 𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
39	4, 38	pm2.61i 182	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∉ wnel 3030  ∀wral 3045  ∃wrex 3054  {crab 3408  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  {cpr 4594  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  Edgcedg 28981  UHGraphcuhgr 28990   NeighbVtx cnbgr 29266

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-edg 28982  df-uhgr 28992  df-nbgr 29267

This theorem is referenced by:  uhgrnbgr0nb  29288