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Theorem nbupgrres 27062
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 = {𝑒𝐸𝑁𝑒}
nbupgrres.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
Assertion
Ref Expression
nbupgrres (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 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
StepHypRef Expression
1 simp1l 1191 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2 eldifi 4106 . . . . . . 7 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾𝑉)
323ad2ant2 1128 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾𝑉)
4 eldifsn 4717 . . . . . . . . 9 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
5 eldifi 4106 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀𝑉)
65anim1i 614 . . . . . . . . 9 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) → (𝑀𝑉𝑀𝐾))
74, 6sylbi 218 . . . . . . . 8 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀𝑉𝑀𝐾))
8 difpr 4734 . . . . . . . 8 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
97, 8eleq2s 2935 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀𝑉𝑀𝐾))
1093ad2ant3 1129 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀𝑉𝑀𝐾))
11 nbupgrres.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
12 nbupgrres.e . . . . . . 7 𝐸 = (Edg‘𝐺)
1311, 12nbupgrel 27043 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐾𝑉) ∧ (𝑀𝑉𝑀𝐾)) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
141, 3, 10, 13syl21anc 835 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
1514biimpa 477 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐸)
168eleq2i 2908 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17 eldifsn 4717 . . . . . . . . . . 11 (𝑀 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑀𝑉𝑀𝑁))
1817anbi1i 623 . . . . . . . . . 10 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
1916, 4, 183bitri 298 . . . . . . . . 9 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
20 simpr 485 . . . . . . . . . . 11 ((𝑀𝑉𝑀𝑁) → 𝑀𝑁)
2120necomd 3075 . . . . . . . . . 10 ((𝑀𝑉𝑀𝑁) → 𝑁𝑀)
2221adantr 481 . . . . . . . . 9 (((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾) → 𝑁𝑀)
2319, 22sylbi 218 . . . . . . . 8 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁𝑀)
24233ad2ant3 1129 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝑀)
25 eldifsn 4717 . . . . . . . . 9 (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾𝑉𝐾𝑁))
26 simpr 485 . . . . . . . . . 10 ((𝐾𝑉𝐾𝑁) → 𝐾𝑁)
2726necomd 3075 . . . . . . . . 9 ((𝐾𝑉𝐾𝑁) → 𝑁𝐾)
2825, 27sylbi 218 . . . . . . . 8 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝐾)
29283ad2ant2 1128 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝐾)
3024, 29nelprd 4592 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31 df-nel 3128 . . . . . 6 (𝑁 ∉ {𝑀, 𝐾} ↔ ¬ 𝑁 ∈ {𝑀, 𝐾})
3230, 31sylibr 235 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ∉ {𝑀, 𝐾})
3332adantr 481 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∉ {𝑀, 𝐾})
34 neleq2 3133 . . . . 5 (𝑒 = {𝑀, 𝐾} → (𝑁𝑒𝑁 ∉ {𝑀, 𝐾}))
35 nbupgrres.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
3634, 35elrab2 3686 . . . 4 ({𝑀, 𝐾} ∈ 𝐹 ↔ ({𝑀, 𝐾} ∈ 𝐸𝑁 ∉ {𝑀, 𝐾}))
3715, 33, 36sylanbrc 583 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐹)
38 nbupgrres.s . . . . . . . 8 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
3911, 12, 35, 38upgrres1 27011 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
40393ad2ant1 1127 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41 simp2 1131 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ (𝑉 ∖ {𝑁}))
4216, 4sylbb 220 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
43423ad2ant3 1129 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
4440, 41, 43jca31 515 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾)))
4544adantr 481 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → ((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾)))
4611, 12, 35, 38upgrres1lem2 27009 . . . . . 6 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
4746eqcomi 2834 . . . . 5 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48 edgval 26750 . . . . . 6 (Edg‘𝑆) = ran (iEdg‘𝑆)
4911, 12, 35, 38upgrres1lem3 27010 . . . . . . 7 (iEdg‘𝑆) = ( I ↾ 𝐹)
5049rneqi 5805 . . . . . 6 ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51 rnresi 5940 . . . . . 6 ran ( I ↾ 𝐹) = 𝐹
5248, 50, 513eqtrri 2853 . . . . 5 𝐹 = (Edg‘𝑆)
5347, 52nbupgrel 27043 . . . 4 (((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾)) → (𝑀 ∈ (𝑆 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐹))
5445, 53syl 17 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → (𝑀 ∈ (𝑆 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐹))
5537, 54mpbird 258 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))
5655ex 413 1 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3020  wnel 3127  {crab 3146  cdif 3936  {csn 4563  {cpr 4565  cop 4569   I cid 5457  ran crn 5554  cres 5555  cfv 6351  (class class class)co 7151  Vtxcvtx 26697  iEdgciedg 26698  Edgcedg 26748  UPGraphcupgr 26781   NeighbVtx cnbgr 27030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-vtx 26699  df-iedg 26700  df-edg 26749  df-upgr 26783  df-nbgr 27031
This theorem is referenced by:  nbupgruvtxres  27105
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