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Theorem nbupgr 28601
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 28593 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
43adantl 483 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
5 simp-4l 782 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → 𝐺 ∈ UPGraph)
6 simpr 486 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒𝐸)
76adantr 482 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → 𝑒𝐸)
8 simpr 486 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → {𝑁, 𝑛} ⊆ 𝑒)
9 simpr 486 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
109adantr 482 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑉)
11 vex 3479 . . . . . . . . . . . 12 𝑛 ∈ V
1211a1i 11 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑛 ∈ V)
13 eldifsn 4791 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑛𝑉𝑛𝑁))
14 simpr 486 . . . . . . . . . . . . . 14 ((𝑛𝑉𝑛𝑁) → 𝑛𝑁)
1514necomd 2997 . . . . . . . . . . . . 13 ((𝑛𝑉𝑛𝑁) → 𝑁𝑛)
1613, 15sylbi 216 . . . . . . . . . . . 12 (𝑛 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝑛)
1716adantl 483 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑛)
1810, 12, 173jca 1129 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
1918adantr 482 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
2019adantr 482 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
211, 2upgredgpr 28402 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑒𝐸 ∧ {𝑁, 𝑛} ⊆ 𝑒) ∧ (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛)) → {𝑁, 𝑛} = 𝑒)
225, 7, 8, 20, 21syl31anc 1374 . . . . . . 7 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → {𝑁, 𝑛} = 𝑒)
2322ex 414 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} = 𝑒))
24 eleq1 2822 . . . . . . 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 3156 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} ∈ 𝐸))
28 simpr 486 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → {𝑁, 𝑛} ∈ 𝐸)
29 sseq2 4009 . . . . . . 7 (𝑒 = {𝑁, 𝑛} → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ {𝑁, 𝑛}))
3029adantl 483 . . . . . 6 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) ∧ 𝑒 = {𝑁, 𝑛}) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ {𝑁, 𝑛}))
31 ssidd 4006 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → {𝑁, 𝑛} ⊆ {𝑁, 𝑛})
3228, 30, 31rspcedvd 3615 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3332ex 414 . . . 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 3440 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
364, 35eqtrd 2773 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wrex 3071  {crab 3433  Vcvv 3475  cdif 3946  wss 3949  {csn 4629  {cpr 4631  cfv 6544  (class class class)co 7409  Vtxcvtx 28256  Edgcedg 28307  UPGraphcupgr 28340   NeighbVtx cnbgr 28589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-edg 28308  df-upgr 28342  df-nbgr 28590
This theorem is referenced by:  nbupgrel  28602  1loopgrnb0  28759
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