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Theorem nbupgruvtxres 26358
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 2651 . . . . . 6 (Vtx‘𝑆) = (Vtx‘𝑆)
21nbgrssovtx 26302 . . . . 5 (𝑆 NeighbVtx 𝐾) ⊆ ((Vtx‘𝑆) ∖ {𝐾})
3 difpr 4366 . . . . . 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 26248 . . . . . . . . 9 (Vtx‘𝑆) = (𝑉 ∖ {𝑁})
98eqcomi 2660 . . . . . . . 8 (𝑉 ∖ {𝑁}) = (Vtx‘𝑆)
109a1i 11 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁}) = (Vtx‘𝑆))
1110difeq1d 3760 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
123, 11syl5eq 2697 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝑉 ∖ {𝑁, 𝐾}) = ((Vtx‘𝑆) ∖ {𝐾}))
132, 12syl5sseqr 3687 . . . 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 472 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})))
1615anim1i 591 . . . . . . 7 (((((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 1056 . . . . . . 7 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) ↔ (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})))
1816, 17sylibr 224 . . . . . 6 (((((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 3924 . . . . . . . . . . . . 13 ((𝑉 ∖ {𝑁}) ∖ {𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
203, 19eqtri 2673 . . . . . . . . . . . 12 (𝑉 ∖ {𝑁, 𝐾}) = ((𝑉 ∖ {𝐾}) ∖ {𝑁})
2120eleq2i 2722 . . . . . . . . . . 11 (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ 𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}))
22 eldifsn 4350 . . . . . . . . . . 11 (𝑛 ∈ ((𝑉 ∖ {𝐾}) ∖ {𝑁}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛𝑁))
2321, 22bitri 264 . . . . . . . . . 10 (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) ↔ (𝑛 ∈ (𝑉 ∖ {𝐾}) ∧ 𝑛𝑁))
2423simplbi 475 . . . . . . . . 9 (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝑉 ∖ {𝐾}))
25 eleq2 2719 . . . . . . . . 9 ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝑛 ∈ (𝑉 ∖ {𝐾})))
2624, 25syl5ibr 236 . . . . . . . 8 ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
2726adantl 481 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾}) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾)))
2827imp 444 . . . . . 6 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝐺 NeighbVtx 𝐾))
294, 5, 6, 7nbupgrres 26310 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁}) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑛 ∈ (𝐺 NeighbVtx 𝐾) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾)))
3018, 28, 29sylc 65 . . . . 5 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})) → 𝑛 ∈ (𝑆 NeighbVtx 𝐾))
3130ralrimiva 2995 . . . 4 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → ∀𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})𝑛 ∈ (𝑆 NeighbVtx 𝐾))
32 dfss3 3625 . . . 4 ((𝑉 ∖ {𝑁, 𝐾}) ⊆ (𝑆 NeighbVtx 𝐾) ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁, 𝐾})𝑛 ∈ (𝑆 NeighbVtx 𝐾))
3331, 32sylibr 224 . . 3 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑉 ∖ {𝑁, 𝐾}) ⊆ (𝑆 NeighbVtx 𝐾))
3414, 33eqssd 3653 . 2 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) ∧ (𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾})) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾}))
3534ex 449 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 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wnel 2926  wral 2941  {crab 2945  cdif 3604  wss 3607  {csn 4210  {cpr 4212  cop 4216   I cid 5052  cres 5145  cfv 5926  (class class class)co 6690  Vtxcvtx 25919  Edgcedg 25984  UPGraphcupgr 26020   NeighbVtx cnbgr 26269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-vtx 25921  df-iedg 25922  df-edg 25985  df-upgr 26022  df-nbgr 26270
This theorem is referenced by:  uvtxupgrres  26359
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