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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 = {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 1254 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐺 ∈ UPGraph)
2 eldifi 3894 . . . . . . 7 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾𝑉)
323ad2ant2 1164 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾𝑉)
4 eldifsn 4472 . . . . . . . . 9 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
5 eldifi 3894 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀𝑉)
65anim1i 608 . . . . . . . . 9 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) → (𝑀𝑉𝑀𝐾))
74, 6sylbi 208 . . . . . . . 8 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀𝑉𝑀𝐾))
8 difpr 4488 . . . . . . . 8 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
97, 8eleq2s 2862 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀𝑉𝑀𝐾))
1093ad2ant3 1165 . . . . . 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 26520 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝐾𝑉) ∧ (𝑀𝑉𝑀𝐾)) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
141, 3, 10, 13syl21anc 866 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) ↔ {𝑀, 𝐾} ∈ 𝐸))
1514biimpa 468 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → {𝑀, 𝐾} ∈ 𝐸)
168eleq2i 2836 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17 eldifsn 4472 . . . . . . . . . . 11 (𝑀 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑀𝑉𝑀𝑁))
1817anbi1i 617 . . . . . . . . . 10 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
1916, 4, 183bitri 288 . . . . . . . . 9 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
20 simpr 477 . . . . . . . . . . 11 ((𝑀𝑉𝑀𝑁) → 𝑀𝑁)
2120necomd 2992 . . . . . . . . . 10 ((𝑀𝑉𝑀𝑁) → 𝑁𝑀)
2221adantr 472 . . . . . . . . 9 (((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾) → 𝑁𝑀)
2319, 22sylbi 208 . . . . . . . 8 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁𝑀)
24233ad2ant3 1165 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝑀)
25 eldifsn 4472 . . . . . . . . 9 (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾𝑉𝐾𝑁))
26 simpr 477 . . . . . . . . . 10 ((𝐾𝑉𝐾𝑁) → 𝐾𝑁)
2726necomd 2992 . . . . . . . . 9 ((𝐾𝑉𝐾𝑁) → 𝑁𝐾)
2825, 27sylbi 208 . . . . . . . 8 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝐾)
29283ad2ant2 1164 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝐾)
3024, 29nelprd 4361 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ¬ 𝑁 ∈ {𝑀, 𝐾})
31 df-nel 3041 . . . . . 6 (𝑁 ∉ {𝑀, 𝐾} ↔ ¬ 𝑁 ∈ {𝑀, 𝐾})
3230, 31sylibr 225 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁 ∉ {𝑀, 𝐾})
3332adantr 472 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁 ∉ {𝑀, 𝐾})
34 neleq2 3046 . . . . 5 (𝑒 = {𝑀, 𝐾} → (𝑁𝑒𝑁 ∉ {𝑀, 𝐾}))
35 nbupgrres.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
3634, 35elrab2 3523 . . . 4 ({𝑀, 𝐾} ∈ 𝐹 ↔ ({𝑀, 𝐾} ∈ 𝐸𝑁 ∉ {𝑀, 𝐾}))
3715, 33, 36sylanbrc 578 . . 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 26484 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑆 ∈ UPGraph)
40393ad2ant1 1163 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑆 ∈ UPGraph)
41 simp2 1167 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾 ∈ (𝑉 ∖ {𝑁}))
4216, 4sylbb 210 . . . . . . 7 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
43423ad2ant3 1165 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
4440, 41, 43jca31 510 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ((𝑆 ∈ UPGraph ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾)))
4544adantr 472 . . . 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 26482 . . . . . 6 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
4746eqcomi 2774 . . . . 5 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48 edgval 26218 . . . . . 6 (Edg‘𝑆) = ran (iEdg‘𝑆)
4911, 12, 35, 38upgrres1lem3 26483 . . . . . . 7 (iEdg‘𝑆) = ( I ↾ 𝐹)
5049rneqi 5520 . . . . . 6 ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51 rnresi 5661 . . . . . 6 ran ( I ↾ 𝐹) = 𝐹
5248, 50, 513eqtrri 2792 . . . . 5 𝐹 = (Edg‘𝑆)
5347, 52nbupgrel 26520 . . . 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 248 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾))
5655ex 401 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 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
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