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Theorem nbupgrres 29381
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 1198 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2 eldifi 4131 . . . . . . 7 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾𝑉)
323ad2ant2 1135 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾𝑉)
4 eldifsn 4786 . . . . . . . . 9 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
5 eldifi 4131 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀𝑉)
65anim1i 615 . . . . . . . . 9 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) → (𝑀𝑉𝑀𝐾))
74, 6sylbi 217 . . . . . . . 8 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀𝑉𝑀𝐾))
8 difpr 4803 . . . . . . . 8 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
97, 8eleq2s 2859 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀𝑉𝑀𝐾))
1093ad2ant3 1136 . . . . . 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 29362 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐾𝑉) ∧ (𝑀𝑉𝑀𝐾)) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
141, 3, 10, 13syl21anc 838 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
1514biimpa 476 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐸)
168eleq2i 2833 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17 eldifsn 4786 . . . . . . . . . . 11 (𝑀 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑀𝑉𝑀𝑁))
1817anbi1i 624 . . . . . . . . . 10 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
1916, 4, 183bitri 297 . . . . . . . . 9 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
20 simpr 484 . . . . . . . . . . 11 ((𝑀𝑉𝑀𝑁) → 𝑀𝑁)
2120necomd 2996 . . . . . . . . . 10 ((𝑀𝑉𝑀𝑁) → 𝑁𝑀)
2221adantr 480 . . . . . . . . 9 (((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾) → 𝑁𝑀)
2319, 22sylbi 217 . . . . . . . 8 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁𝑀)
24233ad2ant3 1136 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝑀)
25 eldifsn 4786 . . . . . . . . 9 (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾𝑉𝐾𝑁))
26 simpr 484 . . . . . . . . . 10 ((𝐾𝑉𝐾𝑁) → 𝐾𝑁)
2726necomd 2996 . . . . . . . . 9 ((𝐾𝑉𝐾𝑁) → 𝑁𝐾)
2825, 27sylbi 217 . . . . . . . 8 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝐾)
29283ad2ant2 1135 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝐾)
3024, 29nelprd 4657 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31 df-nel 3047 . . . . . 6 (𝑁 ∉ {𝑀, 𝐾} ↔ ¬ 𝑁 ∈ {𝑀, 𝐾})
3230, 31sylibr 234 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ∉ {𝑀, 𝐾})
3332adantr 480 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∉ {𝑀, 𝐾})
34 neleq2 3053 . . . . 5 (𝑒 = {𝑀, 𝐾} → (𝑁𝑒𝑁 ∉ {𝑀, 𝐾}))
35 nbupgrres.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
3634, 35elrab2 3695 . . . 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 29330 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
40393ad2ant1 1134 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41 simp2 1138 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ (𝑉 ∖ {𝑁}))
4216, 4sylbb 219 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
43423ad2ant3 1136 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
4440, 41, 43jca31 514 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾)))
4544adantr 480 . . . 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 29328 . . . . . 6 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
4746eqcomi 2746 . . . . 5 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48 edgval 29066 . . . . . 6 (Edg‘𝑆) = ran (iEdg‘𝑆)
4911, 12, 35, 38upgrres1lem3 29329 . . . . . . 7 (iEdg‘𝑆) = ( I ↾ 𝐹)
5049rneqi 5948 . . . . . 6 ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51 rnresi 6093 . . . . . 6 ran ( I ↾ 𝐹) = 𝐹
5248, 50, 513eqtrri 2770 . . . . 5 𝐹 = (Edg‘𝑆)
5347, 52nbupgrel 29362 . . . 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 257 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))
5655ex 412 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  {crab 3436  cdif 3948  {csn 4626  {cpr 4628  cop 4632   I cid 5577  ran crn 5686  cres 5687  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UPGraphcupgr 29097   NeighbVtx cnbgr 29349
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-vtx 29015  df-iedg 29016  df-edg 29065  df-upgr 29099  df-nbgr 29350
This theorem is referenced by:  nbupgruvtxres  29424
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