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Theorem nbupgruvtxres 29424
Description: The neighborhood of a universal vertex in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.) (Proof shortened by AV, 13-Feb-2022.)
Hypotheses
Ref Expression
nbupgruvtxres.v 𝑉 = (Vtx‘𝐺)
nbupgruvtxres.e 𝐸 = (Edg‘𝐺)
nbupgruvtxres.f 𝐹 = {𝑒𝐸𝑁𝑒}
nbupgruvtxres.s 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
Assertion
Ref Expression
nbupgruvtxres (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐺   𝑒,𝐾   𝑒,𝑁   𝑒,𝑉
Allowed substitution hints:   𝑆(𝑒)   𝐹(𝑒)

Proof of Theorem nbupgruvtxres
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (Vtx‘𝑆) = (Vtx‘𝑆)
21nbgrssovtx 29378 . . . . 5 (𝑆 NeighbVtx 𝐾) ⊆ ((Vtx‘𝑆) ∖ {𝐾})
3 difpr 4803 . . . . . 6 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝑁}) ∖ {𝐾})
4 nbupgruvtxres.v . . . . . . . . . 10 𝑉 = (Vtx‘𝐺)
5 nbupgruvtxres.e . . . . . . . . . 10 𝐸 = (Edg‘𝐺)
6 nbupgruvtxres.f . . . . . . . . . 10 𝐹 = {𝑒𝐸𝑁𝑒}
7 nbupgruvtxres.s . . . . . . . . . 10 𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩
84, 5, 6, 7upgrres1lem2 29328 . . . . . . . . 9 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
98eqcomi 2746 . . . . . . . 8 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
109a1i 11 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁}) = (Vtx‘𝑆))
1110difeq1d 4125 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
123, 11eqtrid 2789 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁, 𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
132, 12sseqtrrid 4027 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
1413adantr 480 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) ⊆ (𝑉 ∖ {𝑁, 𝐾}))
15 simpl 482 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})))
1615anim1i 615 . . . . 5 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
17 df-3an 1089 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) ↔ (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
1816, 17sylibr 234 . . . 4 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
19 dif32 4302 . . . . . . . . . . 11 ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
203, 19eqtri 2765 . . . . . . . . . 10 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
2120eleq2i 2833 . . . . . . . . 9 (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}))
22 eldifsn 4786 . . . . . . . . 9 (𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛𝑁))
2321, 22bitri 275 . . . . . . . 8 (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛𝑁))
2423simplbi 497 . . . . . . 7 (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝑉 ∖ {𝐾}))
25 eleq2 2830 . . . . . . 7 ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑛 ∈ (𝑉 ∖ {𝐾})))
2624, 25imbitrrid 246 . . . . . 6 ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
2726adantl 481 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
2827imp 406 . . . 4 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾))
294, 5, 6, 7nbupgrres 29381 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾)))
3018, 28, 29sylc 65 . . 3 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾))
3114, 30eqelssd 4005 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾}))
3231ex 412 1 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wnel 3046  {crab 3436  cdif 3948  wss 3951  {csn 4626  {cpr 4628  cop 4632   I cid 5577  cres 5687  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  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:  uvtxupgrres  29425
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