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Theorem nbupgrres 28021
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 4074 . . . . . . 7 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝐾𝑉)
323ad2ant2 1133 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝐾𝑉)
4 eldifsn 4735 . . . . . . . . 9 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) ↔ (𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾))
5 eldifi 4074 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁}) → 𝑀𝑉)
65anim1i 615 . . . . . . . . 9 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) → (𝑀𝑉𝑀𝐾))
74, 6sylbi 216 . . . . . . . 8 (𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}) → (𝑀𝑉𝑀𝐾))
8 difpr 4751 . . . . . . . 8 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
97, 8eleq2s 2855 . . . . . . 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 28002 . . . . . 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 2828 . . . . . . . . . 10 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑀 ∈ ((𝑉 ∖ {𝑁}) ∖ {𝐾}))
17 eldifsn 4735 . . . . . . . . . . 11 (𝑀 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑀𝑉𝑀𝑁))
1817anbi1i 624 . . . . . . . . . 10 ((𝑀 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀𝐾) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
1916, 4, 183bitri 296 . . . . . . . . 9 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ ((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾))
20 simpr 485 . . . . . . . . . . 11 ((𝑀𝑉𝑀𝑁) → 𝑀𝑁)
2120necomd 2996 . . . . . . . . . 10 ((𝑀𝑉𝑀𝑁) → 𝑁𝑀)
2221adantr 481 . . . . . . . . 9 (((𝑀𝑉𝑀𝑁) ∧ 𝑀𝐾) → 𝑁𝑀)
2319, 22sylbi 216 . . . . . . . 8 (𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑁𝑀)
24233ad2ant3 1134 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝑀)
25 eldifsn 4735 . . . . . . . . 9 (𝐾 ∈ (𝑉 ∖ {𝑁}) ↔ (𝐾𝑉𝐾𝑁))
26 simpr 485 . . . . . . . . . 10 ((𝐾𝑉𝐾𝑁) → 𝐾𝑁)
2726necomd 2996 . . . . . . . . 9 ((𝐾𝑉𝐾𝑁) → 𝑁𝐾)
2825, 27sylbi 216 . . . . . . . 8 (𝐾 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝐾)
29283ad2ant2 1133 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑁𝐾)
3024, 29nelprd 4605 . . . . . 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 233 . . . . 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 3052 . . . . 5 (𝑒 = {𝑀, 𝐾} → (𝑁𝑒𝑁 ∉ {𝑀, 𝐾}))
35 nbupgrres.f . . . . 5 𝐹 = {𝑒𝐸𝑁𝑒}
3634, 35elrab2 3637 . . . 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 27970 . . . . . . 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 218 . . . . . . 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 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 27968 . . . . . 6 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
4746eqcomi 2745 . . . . 5 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
48 edgval 27709 . . . . . 6 (Edg‘𝑆) = ran (iEdg‘𝑆)
4911, 12, 35, 38upgrres1lem3 27969 . . . . . . 7 (iEdg‘𝑆) = ( I ↾ 𝐹)
5049rneqi 5879 . . . . . 6 ran (iEdg‘𝑆) = ran ( I ↾ 𝐹)
51 rnresi 6014 . . . . . 6 ran ( I ↾ 𝐹) = 𝐹
5248, 50, 513eqtrri 2769 . . . . 5 𝐹 = (Edg‘𝑆)
5347, 52nbupgrel 28002 . . . 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 256 . 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 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2940  wnel 3046  {crab 3403  cdif 3895  {csn 4574  {cpr 4576  cop 4580   I cid 5518  ran crn 5622  cres 5623  cfv 6480  (class class class)co 7338  Vtxcvtx 27656  iEdgciedg 27657  Edgcedg 27707  UPGraphcupgr 27740   NeighbVtx cnbgr 27989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-cnex 11029  ax-resscn 11030  ax-1cn 11031  ax-icn 11032  ax-addcl 11033  ax-addrcl 11034  ax-mulcl 11035  ax-mulrcl 11036  ax-mulcom 11037  ax-addass 11038  ax-mulass 11039  ax-distr 11040  ax-i2m1 11041  ax-1ne0 11042  ax-1rid 11043  ax-rnegex 11044  ax-rrecex 11045  ax-cnre 11046  ax-pre-lttri 11047  ax-pre-lttrn 11048  ax-pre-ltadd 11049  ax-pre-mulgt0 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-om 7782  df-1st 7900  df-2nd 7901  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-2o 8369  df-oadd 8372  df-er 8570  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-dju 9759  df-card 9797  df-pnf 11113  df-mnf 11114  df-xr 11115  df-ltxr 11116  df-le 11117  df-sub 11309  df-neg 11310  df-nn 12076  df-2 12138  df-n0 12336  df-xnn0 12408  df-z 12422  df-uz 12685  df-fz 13342  df-hash 14147  df-vtx 27658  df-iedg 27659  df-edg 27708  df-upgr 27742  df-nbgr 27990
This theorem is referenced by:  nbupgruvtxres  28064
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