Theorem nbupgrres 27731

Description: The neighborhood of a vertex in a restricted pseudograph (not necessarily valid for a hypergraph, because 𝑁, 𝐾 and 𝑀 could be connected by one edge, so 𝑀 is a neighbor of 𝐾 in the original graph, but not in the restricted graph, because the edge between 𝑀 and 𝐾, also incident with 𝑁, was removed). (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)

Hypotheses

Ref	Expression
nbupgrres.v	⊢ 𝑉 = (Vtx‘𝐺)
nbupgrres.e	⊢ 𝐸 = (Edg‘𝐺)
nbupgrres.f	⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
nbupgrres.s	⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩

Assertion

Ref	Expression
nbupgrres	⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾)))

Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝐾   𝑒,𝑁   𝑒,𝑀   𝑒,𝑉

Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem nbupgrres

Step	Hyp	Ref	Expression
1		simp1l 1196	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2		eldifi 4061	. . . . . . 7 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾 ∈ 𝑉)
3	2	3ad2ant2 1133	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ 𝑉)
4		eldifsn 4720	. . . . . . . . 9 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
5		eldifi 4061	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀 ∈ 𝑉)
6	5	anim1i 615	. . . . . . . . 9 ⊢ ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
7	4, 6	sylbi 216	. . . . . . . 8 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
8		difpr 4736	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
9	7, 8	eleq2s 2857	. . . . . . 7 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
10	9	3ad2ant3 1134	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
11		nbupgrres.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
12		nbupgrres.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
13	11, 12	nbupgrel 27712	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾)) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
14	1, 3, 10, 13	syl21anc 835	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
15	14	biimpa 477	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐸)
16	8	eleq2i 2830	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17		eldifsn 4720	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁))
18	17	anbi1i 624	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾) ↔ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾))
19	16, 4, 18	3bitri 297	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾))
20		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
21	20	necomd 2999	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑁 ≠ 𝑀)
22	21	adantr 481	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾) → 𝑁 ≠ 𝑀)
23	19, 22	sylbi 216	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁 ≠ 𝑀)
24	23	3ad2ant3 1134	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ≠ 𝑀)
25		eldifsn 4720	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁))
26		simpr 485	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝐾 ≠ 𝑁)
27	26	necomd 2999	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝑁 ≠ 𝐾)
28	25, 27	sylbi 216	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ≠ 𝐾)
29	28	3ad2ant2 1133	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ≠ 𝐾)
30	24, 29	nelprd 4592	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31		df-nel 3050	. . . . . 6 ⊢ (𝑁 ∉ {𝑀, 𝐾} ↔ ¬ 𝑁 ∈ {𝑀, 𝐾})
32	30, 31	sylibr 233	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ∉ {𝑀, 𝐾})
33	32	adantr 481	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∉ {𝑀, 𝐾})
34		neleq2 3055	. . . . 5 ⊢ (𝑒 = {𝑀, 𝐾} → (𝑁 ∉ 𝑒 ↔ 𝑁 ∉ {𝑀, 𝐾}))
35		nbupgrres.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
36	34, 35	elrab2 3627	. . . 4 ⊢ ({𝑀, 𝐾} ∈ 𝐹 ↔ ({𝑀, 𝐾} ∈ 𝐸 ∧ 𝑁 ∉ {𝑀, 𝐾}))
37	15, 33, 36	sylanbrc 583	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐹)
38		nbupgrres.s	. . . . . . . 8 ⊢ 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
39	11, 12, 35, 38	upgrres1 27680	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)
40	39	3ad2ant1 1132	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41		simp2 1136	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ (𝑉 ∖ {𝑁}))
42	16, 4	sylbb 218	. . . . . . 7 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
43	42	3ad2ant3 1134	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
44	40, 41, 43	jca31 515	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾)))
45	44	adantr 481	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → ((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾)))
46	11, 12, 35, 38	upgrres1lem2 27678	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
47	46	eqcomi 2747	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48		edgval 27419	. . . . . 6 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
49	11, 12, 35, 38	upgrres1lem3 27679	. . . . . . 7 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
50	49	rneqi 5846	. . . . . 6 ⊢ ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51		rnresi 5983	. . . . . 6 ⊢ ran ( I ↾ 𝐹) = 𝐹
52	48, 50, 51	3eqtrri 2771	. . . . 5 ⊢ 𝐹 = (Edg‘𝑆)
53	47, 52	nbupgrel 27712	. . . 4 ⊢ (((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾)) → (𝑀 ∈ (𝑆 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐹))
54	45, 53	syl 17	. . 3 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → (𝑀 ∈ (𝑆 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐹))
55	37, 54	mpbird 256	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))
56	55	ex 413	1 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∉ wnel 3049  {crab 3068   ∖ cdif 3884  {csn 4561  {cpr 4563  ⟨cop 4567   I cid 5488  ran crn 5590   ↾ cres 5591  ‘cfv 6433  (class class class)co 7275  Vtxcvtx 27366  iEdgciedg 27367  Edgcedg 27417  UPGraphcupgr 27450   NeighbVtx cnbgr 27699

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-vtx 27368  df-iedg 27369  df-edg 27418  df-upgr 27452  df-nbgr 27700

This theorem is referenced by:  nbupgruvtxres  27774