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Theorem nbupgrres 29395
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 1196 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2 eldifi 4140 . . . . . . 7 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾𝑉)
323ad2ant2 1133 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾𝑉)
4 eldifsn 4790 . . . . . . . . 9 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
5 eldifi 4140 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀𝑉)
65anim1i 615 . . . . . . . . 9 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) → (𝑀𝑉𝑀𝐾))
74, 6sylbi 217 . . . . . . . 8 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀𝑉𝑀𝐾))
8 difpr 4807 . . . . . . . 8 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
97, 8eleq2s 2856 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀𝑉𝑀𝐾))
1093ad2ant3 1134 . . . . . 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 29376 . . . . . 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 2830 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17 eldifsn 4790 . . . . . . . . . . 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 2993 . . . . . . . . . 10 ((𝑀𝑉𝑀𝑁) → 𝑁𝑀)
2221adantr 480 . . . . . . . . 9 (((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾) → 𝑁𝑀)
2319, 22sylbi 217 . . . . . . . 8 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁𝑀)
24233ad2ant3 1134 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝑀)
25 eldifsn 4790 . . . . . . . . 9 (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾𝑉𝐾𝑁))
26 simpr 484 . . . . . . . . . 10 ((𝐾𝑉𝐾𝑁) → 𝐾𝑁)
2726necomd 2993 . . . . . . . . 9 ((𝐾𝑉𝐾𝑁) → 𝑁𝐾)
2825, 27sylbi 217 . . . . . . . 8 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝐾)
29283ad2ant2 1133 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝐾)
3024, 29nelprd 4661 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31 df-nel 3044 . . . . . 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 3050 . . . . 5 (𝑒 = {𝑀, 𝐾} → (𝑁𝑒𝑁 ∉ {𝑀, 𝐾}))
35 nbupgrres.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
3634, 35elrab2 3697 . . . 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 29344 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
40393ad2ant1 1132 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41 simp2 1136 . . . . . 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 1134 . . . . . 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 29342 . . . . . 6 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
4746eqcomi 2743 . . . . 5 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48 edgval 29080 . . . . . 6 (Edg‘𝑆) = ran (iEdg‘𝑆)
4911, 12, 35, 38upgrres1lem3 29343 . . . . . . 7 (iEdg‘𝑆) = ( I ↾ 𝐹)
5049rneqi 5950 . . . . . 6 ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51 rnresi 6094 . . . . . 6 ran ( I ↾ 𝐹) = 𝐹
5248, 50, 513eqtrri 2767 . . . . 5 𝐹 = (Edg‘𝑆)
5347, 52nbupgrel 29376 . . . 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 1086   = wceq 1536  wcel 2105  wne 2937  wnel 3043  {crab 3432  cdif 3959  {csn 4630  {cpr 4632  cop 4636   I cid 5581  ran crn 5689  cres 5690  cfv 6562  (class class class)co 7430  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UPGraphcupgr 29111   NeighbVtx cnbgr 29363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366  df-vtx 29029  df-iedg 29030  df-edg 29079  df-upgr 29113  df-nbgr 29364
This theorem is referenced by:  nbupgruvtxres  29438
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