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Theorem nbupgr 28856
Description: The set of neighbors of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.)
Hypotheses
Ref Expression
nbuhgr.v 𝑉 = (Vtx‘𝐺)
nbuhgr.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbupgr ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
Distinct variable groups:   𝑛,𝐺   𝑛,𝑁   𝑛,𝑉   𝑛,𝐸

Proof of Theorem nbupgr
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 nbuhgr.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbuhgr.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 28848 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
43adantl 482 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
5 simp-4l 781 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → 𝐺 ∈ UPGraph)
6 simpr 485 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒𝐸)
76adantr 481 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → 𝑒𝐸)
8 simpr 485 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → {𝑁, 𝑛} ⊆ 𝑒)
9 simpr 485 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
109adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑉)
11 vex 3478 . . . . . . . . . . . 12 𝑛 ∈ V
1211a1i 11 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑛 ∈ V)
13 eldifsn 4790 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑛𝑉𝑛𝑁))
14 simpr 485 . . . . . . . . . . . . . 14 ((𝑛𝑉𝑛𝑁) → 𝑛𝑁)
1514necomd 2996 . . . . . . . . . . . . 13 ((𝑛𝑉𝑛𝑁) → 𝑁𝑛)
1613, 15sylbi 216 . . . . . . . . . . . 12 (𝑛 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝑛)
1716adantl 482 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑛)
1810, 12, 173jca 1128 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
1918adantr 481 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
2019adantr 481 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
211, 2upgredgpr 28657 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑒𝐸 ∧ {𝑁, 𝑛} ⊆ 𝑒) ∧ (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛)) → {𝑁, 𝑛} = 𝑒)
225, 7, 8, 20, 21syl31anc 1373 . . . . . . 7 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → {𝑁, 𝑛} = 𝑒)
2322ex 413 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} = 𝑒))
24 eleq1 2821 . . . . . . 7 ({𝑁, 𝑛} = 𝑒 → ({𝑁, 𝑛} ∈ 𝐸𝑒𝐸))
2524biimprd 247 . . . . . 6 ({𝑁, 𝑛} = 𝑒 → (𝑒𝐸 → {𝑁, 𝑛} ∈ 𝐸))
2623, 6, 25syl6ci 71 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} ∈ 𝐸))
2726rexlimdva 3155 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} ∈ 𝐸))
28 simpr 485 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → {𝑁, 𝑛} ∈ 𝐸)
29 sseq2 4008 . . . . . . 7 (𝑒 = {𝑁, 𝑛} → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ {𝑁, 𝑛}))
3029adantl 482 . . . . . 6 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) ∧ 𝑒 = {𝑁, 𝑛}) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ {𝑁, 𝑛}))
31 ssidd 4005 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → {𝑁, 𝑛} ⊆ {𝑁, 𝑛})
3228, 30, 31rspcedvd 3614 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3332ex 413 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ({𝑁, 𝑛} ∈ 𝐸 → ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3427, 33impbid 211 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ∈ 𝐸))
3534rabbidva 3439 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
364, 35eqtrd 2772 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070  {crab 3432  Vcvv 3474  cdif 3945  wss 3948  {csn 4628  {cpr 4630  cfv 6543  (class class class)co 7411  Vtxcvtx 28511  Edgcedg 28562  UPGraphcupgr 28595   NeighbVtx cnbgr 28844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-hash 14295  df-edg 28563  df-upgr 28597  df-nbgr 28845
This theorem is referenced by:  nbupgrel  28857  1loopgrnb0  29014
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