Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nbupgr Structured version   Visualization version   GIF version

Theorem nbupgr 40566
 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
nbgrel.v 𝑉 = (Vtx‘𝐺)
nbgrel.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 nbgrel.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbgrel.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 40560 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
43adantl 481 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
5 simp-4l 802 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → 𝐺 ∈ UPGraph )
6 simpr 476 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒𝐸)
76adantr 480 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → 𝑒𝐸)
8 simpr 476 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → {𝑁, 𝑛} ⊆ 𝑒)
9 simpr 476 . . . . . . . . . . . 12 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → 𝑁𝑉)
109adantr 480 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑉)
11 vex 3176 . . . . . . . . . . . 12 𝑛 ∈ V
1211a1i 11 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑛 ∈ V)
13 eldifsn 4260 . . . . . . . . . . . . 13 (𝑛 ∈ (𝑉 ∖ {𝑁}) ↔ (𝑛𝑉𝑛𝑁))
14 simpr 476 . . . . . . . . . . . . . 14 ((𝑛𝑉𝑛𝑁) → 𝑛𝑁)
1514necomd 2837 . . . . . . . . . . . . 13 ((𝑛𝑉𝑛𝑁) → 𝑁𝑛)
1613, 15sylbi 206 . . . . . . . . . . . 12 (𝑛 ∈ (𝑉 ∖ {𝑁}) → 𝑁𝑛)
1716adantl 481 . . . . . . . . . . 11 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝑁𝑛)
1810, 12, 173jca 1235 . . . . . . . . . 10 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
1918adantr 480 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
2019adantr 480 . . . . . . . 8 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛))
211, 2upgredgpr 25815 . . . . . . . 8 (((𝐺 ∈ UPGraph ∧ 𝑒𝐸 ∧ {𝑁, 𝑛} ⊆ 𝑒) ∧ (𝑁𝑉𝑛 ∈ V ∧ 𝑁𝑛)) → {𝑁, 𝑛} = 𝑒)
225, 7, 8, 20, 21syl31anc 1321 . . . . . . 7 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) ∧ {𝑁, 𝑛} ⊆ 𝑒) → {𝑁, 𝑛} = 𝑒)
2322ex 449 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} = 𝑒))
24 eleq1 2676 . . . . . . 7 ({𝑁, 𝑛} = 𝑒 → ({𝑁, 𝑛} ∈ 𝐸𝑒𝐸))
2524biimprd 237 . . . . . 6 ({𝑁, 𝑛} = 𝑒 → (𝑒𝐸 → {𝑁, 𝑛} ∈ 𝐸))
2623, 6, 25syl6ci 69 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} ∈ 𝐸))
2726rexlimdva 3013 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 → {𝑁, 𝑛} ∈ 𝐸))
28 simpr 476 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → {𝑁, 𝑛} ∈ 𝐸)
29 sseq2 3590 . . . . . . 7 (𝑒 = {𝑁, 𝑛} → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ {𝑁, 𝑛}))
3029adantl 481 . . . . . 6 (((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) ∧ 𝑒 = {𝑁, 𝑛}) → ({𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ⊆ {𝑁, 𝑛}))
31 ssid 3587 . . . . . . 7 {𝑁, 𝑛} ⊆ {𝑁, 𝑛}
3231a1i 11 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → {𝑁, 𝑛} ⊆ {𝑁, 𝑛})
3328, 30, 32rspcedvd 3289 . . . . 5 ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ {𝑁, 𝑛} ∈ 𝐸) → ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3433ex 449 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ({𝑁, 𝑛} ∈ 𝐸 → ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3527, 34impbid 201 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒 ↔ {𝑁, 𝑛} ∈ 𝐸))
3635rabbidva 3163 . 2 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
374, 36eqtrd 2644 1 ((𝐺 ∈ UPGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ {𝑁, 𝑛} ∈ 𝐸})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673   UPGraph cupgr 25747  Edgcedga 25792   NeighbVtx cnbgr 40550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-edga 25793  df-nbgr 40554 This theorem is referenced by:  nbupgrel  40567  1loopgrnb0  40717
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