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Theorem hashnbusgrvd 27003
 Description: In a simple graph, the number of neighbors of a vertex is the degree of this vertex. This theorem does not hold for (simple) pseudographs, because a vertex connected with itself only by a loop has no neighbors, see uspgrloopnb0 26994, but degree 2, see uspgrloopvd2 26995. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 27001, but also degree 2, see umgr2v2evd2 27002. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
Hypothesis
Ref Expression
hashnbusgrvd.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
hashnbusgrvd ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))

Proof of Theorem hashnbusgrvd
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 hashnbusgrvd.v . . 3 𝑉 = (Vtx‘𝐺)
2 eqid 2772 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbedgusgr 26847 . 2 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑈𝑒}))
4 eqid 2772 . . 3 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
51, 2, 4vtxdusgrfvedg 26966 . 2 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑈𝑒}))
63, 5eqtr4d 2811 1 ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2048  {crab 3086  ‘cfv 6182  (class class class)co 6970  ♯chash 13498  Vtxcvtx 26474  Edgcedg 26525  USGraphcusgr 26627   NeighbVtx cnbgr 26807  VtxDegcvtxdg 26940 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-dju 9116  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-n0 11701  df-xnn0 11773  df-z 11787  df-uz 12052  df-xadd 12318  df-fz 12702  df-hash 13499  df-edg 26526  df-uhgr 26536  df-ushgr 26537  df-upgr 26560  df-umgr 26561  df-uspgr 26628  df-usgr 26629  df-nbgr 26808  df-vtxdg 26941 This theorem is referenced by:  usgruvtxvdb  27004  vtxdusgradjvtx  27007  cusgrrusgr  27056  rusgrpropnb  27058  frgrncvvdeq  27833  fusgreghash2wspv  27859
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