Theorem hashnbusgrvd 27304

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 27295, but degree 2, see uspgrloopvd2 27296. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 27302, but also degree 2, see umgr2v2evd2 27303. (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 ∧ 𝑈 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))

Proof of Theorem hashnbusgrvd

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashnbusgrvd.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2821	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	nbedgusgr 27148	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑈 ∈ 𝑒}))
4		eqid 2821	. . 3 ⊢ (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
5	1, 2, 4	vtxdusgrfvedg 27267	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → ((VtxDeg‘𝐺)‘𝑈) = (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑈 ∈ 𝑒}))
6	3, 5	eqtr4d 2859	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑈 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {crab 3142  ‘cfv 6350  (class class class)co 7150  ♯chash 13684  Vtxcvtx 26775  Edgcedg 26826  USGraphcusgr 26928   NeighbVtx cnbgr 27108  VtxDegcvtxdg 27241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-xadd 12502  df-fz 12887  df-hash 13685  df-edg 26827  df-uhgr 26837  df-ushgr 26838  df-upgr 26861  df-umgr 26862  df-uspgr 26929  df-usgr 26930  df-nbgr 27109  df-vtxdg 27242

This theorem is referenced by:  usgruvtxvdb  27305  vtxdusgradjvtx  27308  cusgrrusgr  27357  rusgrpropnb  27359  frgrncvvdeq  28082  fusgreghash2wspv  28108