Theorem nbupgrres 26544

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 1254	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2		eldifi 3894	. . . . . . 7 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾 ∈ 𝑉)
3	2	3ad2ant2 1164	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ 𝑉)
4		eldifsn 4472	. . . . . . . . 9 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
5		eldifi 3894	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀 ∈ 𝑉)
6	5	anim1i 608	. . . . . . . . 9 ⊢ ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
7	4, 6	sylbi 208	. . . . . . . 8 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
8		difpr 4488	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
9	7, 8	eleq2s 2862	. . . . . . 7 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾))
10	9	3ad2ant3 1165	. . . . . 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 26520	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐾 ∈ 𝑉) ∧ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝐾)) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
14	1, 3, 10, 13	syl21anc 866	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
15	14	biimpa 468	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐸)
16	8	eleq2i 2836	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17		eldifsn 4472	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁))
18	17	anbi1i 617	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾) ↔ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾))
19	16, 4, 18	3bitri 288	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾))
20		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ≠ 𝑁)
21	20	necomd 2992	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑁 ≠ 𝑀)
22	21	adantr 472	. . . . . . . . 9 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) ∧ 𝑀 ≠ 𝐾) → 𝑁 ≠ 𝑀)
23	19, 22	sylbi 208	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁 ≠ 𝑀)
24	23	3ad2ant3 1165	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ≠ 𝑀)
25		eldifsn 4472	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁))
26		simpr 477	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝐾 ≠ 𝑁)
27	26	necomd 2992	. . . . . . . . 9 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝑁 ≠ 𝐾)
28	25, 27	sylbi 208	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁 ≠ 𝐾)
29	28	3ad2ant2 1164	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ≠ 𝐾)
30	24, 29	nelprd 4361	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31		df-nel 3041	. . . . . 6 ⊢ (𝑁 ∉ {𝑀, 𝐾} ↔ ¬ 𝑁 ∈ {𝑀, 𝐾})
32	30, 31	sylibr 225	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ∉ {𝑀, 𝐾})
33	32	adantr 472	. . . 4 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∉ {𝑀, 𝐾})
34		neleq2 3046	. . . . 5 ⊢ (𝑒 = {𝑀, 𝐾} → (𝑁 ∉ 𝑒 ↔ 𝑁 ∉ {𝑀, 𝐾}))
35		nbupgrres.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
36	34, 35	elrab2 3523	. . . 4 ⊢ ({𝑀, 𝐾} ∈ 𝐹 ↔ ({𝑀, 𝐾} ∈ 𝐸 ∧ 𝑁 ∉ {𝑀, 𝐾}))
37	15, 33, 36	sylanbrc 578	. . 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 26484	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) → 𝑆 ∈ UPGraph)
40	39	3ad2ant1 1163	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41		simp2 1167	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ (𝑉 ∖ {𝑁}))
42	16, 4	sylbb 210	. . . . . . 7 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
43	42	3ad2ant3 1165	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
44	40, 41, 43	jca31 510	. . . . 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 472	. . . 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 26482	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
47	46	eqcomi 2774	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48		edgval 26218	. . . . . 6 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
49	11, 12, 35, 38	upgrres1lem3 26483	. . . . . . 7 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
50	49	rneqi 5520	. . . . . 6 ⊢ ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51		rnresi 5661	. . . . . 6 ⊢ ran ( I ↾ 𝐹) = 𝐹
52	48, 50, 51	3eqtrri 2792	. . . . 5 ⊢ 𝐹 = (Edg‘𝑆)
53	47, 52	nbupgrel 26520	. . . 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 248	. 2 ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))
56	55	ex 401	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 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155   ≠ wne 2937   ∉ wnel 3040  {crab 3059   ∖ cdif 3729  {csn 4334  {cpr 4336  ⟨cop 4340   I cid 5184  ran crn 5278   ↾ cres 5279  ‘cfv 6068  (class class class)co 6842  Vtxcvtx 26165  iEdgciedg 26166  Edgcedg 26216  UPGraphcupgr 26252   NeighbVtx cnbgr 26503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-xnn0 11611  df-z 11625  df-uz 11887  df-fz 12534  df-hash 13322  df-vtx 26167  df-iedg 26168  df-edg 26217  df-upgr 26254  df-nbgr 26504

This theorem is referenced by:  nbupgruvtxres  26593