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

Theorem frgrncvvdeq 41480
 Description: In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrncvvdeq.v 𝑉 = (Vtx‘𝐺)
frgrncvvdeq.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
frgrncvvdeq (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑉,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑦)

Proof of Theorem frgrncvvdeq
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . . . . . . 7 (𝐺 NeighbVtx 𝑥) ∈ V
21a1i 11 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 NeighbVtx 𝑥) ∈ V)
3 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
4 eqid 2610 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
5 eqid 2610 . . . . . . 7 (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
6 eqid 2610 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
7 simpl 472 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑉)
87ad2antlr 759 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑉)
9 eldifi 3694 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦𝑉)
109adantl 481 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦𝑉)
1110ad2antlr 759 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦𝑉)
12 eldif 3550 . . . . . . . . . 10 (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
13 velsn 4141 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
1413biimpri 217 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 ∈ {𝑥})
1514equcoms 1934 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 ∈ {𝑥})
1615necon3bi 2808 . . . . . . . . . 10 𝑦 ∈ {𝑥} → 𝑥𝑦)
1712, 16simplbiim 657 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑦)
1817adantl 481 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑦)
1918ad2antlr 759 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑦)
20 simpr 476 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
21 simpl 472 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
2221adantr 480 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
23 eqid 2610 . . . . . . 7 (𝑎 ∈ (𝐺 NeighbVtx 𝑥) ↦ (𝑏 ∈ (𝐺 NeighbVtx 𝑦){𝑎, 𝑏} ∈ (Edg‘𝐺))) = (𝑎 ∈ (𝐺 NeighbVtx 𝑥) ↦ (𝑏 ∈ (𝐺 NeighbVtx 𝑦){𝑎, 𝑏} ∈ (Edg‘𝐺)))
243, 4, 5, 6, 8, 11, 19, 20, 22, 23frgrncvvdeqlem8 41479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝑎 ∈ (𝐺 NeighbVtx 𝑥) ↦ (𝑏 ∈ (𝐺 NeighbVtx 𝑦){𝑎, 𝑏} ∈ (Edg‘𝐺))):(𝐺 NeighbVtx 𝑥)–1-1-onto→(𝐺 NeighbVtx 𝑦))
252, 24hasheqf1od 13006 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑥)) = (#‘(𝐺 NeighbVtx 𝑦)))
26 frgrusgr 41432 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
2726, 7anim12i 588 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
2827adantr 480 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
293hashnbusgrvd 40744 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑥𝑉) → (#‘(𝐺 NeighbVtx 𝑥)) = ((VtxDeg‘𝐺)‘𝑥))
3028, 29syl 17 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑥)) = ((VtxDeg‘𝐺)‘𝑥))
3126, 10anim12i 588 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
3231adantr 480 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
333hashnbusgrvd 40744 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑦𝑉) → (#‘(𝐺 NeighbVtx 𝑦)) = ((VtxDeg‘𝐺)‘𝑦))
3432, 33syl 17 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑦)) = ((VtxDeg‘𝐺)‘𝑦))
3525, 30, 343eqtr3d 2652 . . . 4 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘𝑦))
36 frgrncvvdeq.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
3736fveq1i 6104 . . . 4 (𝐷𝑥) = ((VtxDeg‘𝐺)‘𝑥)
3836fveq1i 6104 . . . 4 (𝐷𝑦) = ((VtxDeg‘𝐺)‘𝑦)
3935, 37, 383eqtr4g 2669 . . 3 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷𝑥) = (𝐷𝑦))
4039ex 449 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
4140ralrimivva 2954 1 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  {csn 4125  {cpr 4127   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  #chash 12979  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550  VtxDegcvtxdg 40681   FriendGraph cfrgr 41428 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-fal 1481  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-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-nbgr 40554  df-vtxdg 40682  df-frgr 41429 This theorem is referenced by:  frgrwopreglem4  41484
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