Theorem nbupgrres 29309

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 4082	. . . . . . 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 4737	. . . . . . . . 9 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
5		eldifi 4082	. . . . . . . . . 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 4754	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
9	7, 8	eleq2s 2846	. . . . . . 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 29290	. . . . . 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 2820	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17		eldifsn 4737	. . . . . . . . . . 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 2980	. . . . . . . . . 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 4737	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁))
26		simpr 484	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝐾 ≠ 𝑁)
27	26	necomd 2980	. . . . . . . . 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 4609	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31		df-nel 3030	. . . . . 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 3036	. . . . 5 ⊢ (𝑒 = {𝑀, 𝐾} → (𝑁 ∉ 𝑒 ↔ 𝑁 ∉ {𝑀, 𝐾}))
35		nbupgrres.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
36	34, 35	elrab2 3651	. . . 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 29258	. . . . . . 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 29256	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
47	46	eqcomi 2738	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48		edgval 28994	. . . . . 6 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
49	11, 12, 35, 38	upgrres1lem3 29257	. . . . . . 7 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
50	49	rneqi 5879	. . . . . 6 ⊢ ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51		rnresi 6026	. . . . . 6 ⊢ ran ( I ↾ 𝐹) = 𝐹
52	48, 50, 51	3eqtrri 2757	. . . . 5 ⊢ 𝐹 = (Edg‘𝑆)
53	47, 52	nbupgrel 29290	. . . 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 2925   ∉ wnel 3029  {crab 3394   ∖ cdif 3900  {csn 4577  {cpr 4579  ⟨cop 4583   I cid 5513  ran crn 5620   ↾ cres 5621  ‘cfv 6482  (class class class)co 7349  Vtxcvtx 28941  iEdgciedg 28942  Edgcedg 28992  UPGraphcupgr 29025   NeighbVtx cnbgr 29277

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 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-fz 13411  df-hash 14238  df-vtx 28943  df-iedg 28944  df-edg 28993  df-upgr 29027  df-nbgr 29278

This theorem is referenced by:  nbupgruvtxres  29352