Theorem nbupgrres 29343

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 4106	. . . . . . 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 4762	. . . . . . . . 9 ⊢ (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ≠ 𝐾))
5		eldifi 4106	. . . . . . . . . 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 4779	. . . . . . . 8 ⊢ (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
9	7, 8	eleq2s 2852	. . . . . . 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 29324	. . . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17		eldifsn 4762	. . . . . . . . . . 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 2987	. . . . . . . . . 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 4762	. . . . . . . . 9 ⊢ (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁))
26		simpr 484	. . . . . . . . . 10 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐾 ≠ 𝑁) → 𝐾 ≠ 𝑁)
27	26	necomd 2987	. . . . . . . . 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 4633	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31		df-nel 3037	. . . . . 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 3043	. . . . 5 ⊢ (𝑒 = {𝑀, 𝐾} → (𝑁 ∉ 𝑒 ↔ 𝑁 ∉ {𝑀, 𝐾}))
35		nbupgrres.f	. . . . 5 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒}
36	34, 35	elrab2 3674	. . . 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 29292	. . . . . . 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 29290	. . . . . 6 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
47	46	eqcomi 2744	. . . . 5 ⊢ (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48		edgval 29028	. . . . . 6 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
49	11, 12, 35, 38	upgrres1lem3 29291	. . . . . . 7 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹)
50	49	rneqi 5917	. . . . . 6 ⊢ ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51		rnresi 6062	. . . . . 6 ⊢ ran ( I ↾ 𝐹) = 𝐹
52	48, 50, 51	3eqtrri 2763	. . . . 5 ⊢ 𝐹 = (Edg‘𝑆)
53	47, 52	nbupgrel 29324	. . . 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 2108   ≠ wne 2932   ∉ wnel 3036  {crab 3415   ∖ cdif 3923  {csn 4601  {cpr 4603  ⟨cop 4607   I cid 5547  ran crn 5655   ↾ cres 5656  ‘cfv 6531  (class class class)co 7405  Vtxcvtx 28975  iEdgciedg 28976  Edgcedg 29026  UPGraphcupgr 29059   NeighbVtx cnbgr 29311

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-n0 12502  df-xnn0 12575  df-z 12589  df-uz 12853  df-fz 13525  df-hash 14349  df-vtx 28977  df-iedg 28978  df-edg 29027  df-upgr 29061  df-nbgr 29312

This theorem is referenced by:  nbupgruvtxres  29386