Theorem nbupgrres 29297

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 1198	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2		eldifi 4096	. . . . . . 7 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾 ∈ 𝑉)
3	2	3ad2ant2 1134	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ 𝑉)
4		eldifsn 4752	. . . . . . . . 9 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
5		eldifi 4096	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀 ∈ 𝑉)
6	5	anim1i 615	. . . . . . . . 9 ⊢ ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
7	4, 6	sylbi 217	. . . . . . . 8 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
8		difpr 4769	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
9	7, 8	eleq2s 2847	. . . . . . 7 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
10	9	3ad2ant3 1135	. . . . . 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 29278	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾)) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
14	1, 3, 10, 13	syl21anc 837	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
15	14	biimpa 476	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐸)
16	8	eleq2i 2821	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17		eldifsn 4752	. . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
21	20	necomd 2981	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑁 ≠ 𝑀)
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾) → 𝑁 ≠ 𝑀)
23	19, 22	sylbi 217	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁 ≠ 𝑀)
24	23	3ad2ant3 1135	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ≠ 𝑀)
25		eldifsn 4752	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁))
26		simpr 484	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝐾 ≠ 𝑁)
27	26	necomd 2981	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝑁 ≠ 𝐾)
28	25, 27	sylbi 217	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ≠ 𝐾)
29	28	3ad2ant2 1134	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ≠ 𝐾)
30	24, 29	nelprd 4623	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31		df-nel 3031	. . . . . 6 ⊢ (𝑁 ∉ {𝑀, 𝐾} ↔ ¬ 𝑁 ∈ {𝑀, 𝐾})
32	30, 31	sylibr 234	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ∉ {𝑀, 𝐾})
33	32	adantr 480	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∉ {𝑀, 𝐾})
34		neleq2 3037	. . . . 5 ⊢ (𝑒 = {𝑀, 𝐾} → (𝑁 ∉ 𝑒 ↔ 𝑁 ∉ {𝑀, 𝐾}))
35		nbupgrres.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
36	34, 35	elrab2 3664	. . . 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 29246	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)
40	39	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41		simp2 1137	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ (𝑉 ∖ {𝑁}))
42	16, 4	sylbb 219	. . . . . . 7 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
43	42	3ad2ant3 1135	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
44	40, 41, 43	jca31 514	. . . . 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 480	. . . 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 29244	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
47	46	eqcomi 2739	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48		edgval 28982	. . . . . 6 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
49	11, 12, 35, 38	upgrres1lem3 29245	. . . . . . 7 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
50	49	rneqi 5903	. . . . . 6 ⊢ ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51		rnresi 6048	. . . . . 6 ⊢ ran ( I ↾ 𝐹) = 𝐹
52	48, 50, 51	3eqtrri 2758	. . . . 5 ⊢ 𝐹 = (Edg‘𝑆)
53	47, 52	nbupgrel 29278	. . . 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 257	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))
56	55	ex 412	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   ∉ wnel 3030  {crab 3408   ∖ cdif 3913  {csn 4591  {cpr 4593  ⟨cop 4597   I cid 5534  ran crn 5641   ↾ cres 5642  ‘cfv 6513  (class class class)co 7389  Vtxcvtx 28929  iEdgciedg 28930  Edgcedg 28980  UPGraphcupgr 29013   NeighbVtx cnbgr 29265

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 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-om 7845  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-oadd 8440  df-er 8673  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-dju 9860  df-card 9898  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-n0 12449  df-xnn0 12522  df-z 12536  df-uz 12800  df-fz 13475  df-hash 14302  df-vtx 28931  df-iedg 28932  df-edg 28981  df-upgr 29015  df-nbgr 29266

This theorem is referenced by:  nbupgruvtxres  29340