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Theorem frgrncvvdeq 27289
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 ovexd 6720 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 NeighbVtx 𝑥) ∈ V)
2 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
3 eqid 2651 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
4 eqid 2651 . . . . . . 7 (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5 eqid 2651 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6 simpl 472 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑉)
76ad2antlr 763 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑉)
8 eldifi 3765 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦𝑉)
98adantl 481 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦𝑉)
109ad2antlr 763 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦𝑉)
11 eldif 3617 . . . . . . . . . 10 (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12 velsn 4226 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
1312biimpri 218 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 ∈ {𝑥})
1413equcoms 1993 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 ∈ {𝑥})
1514necon3bi 2849 . . . . . . . . . 10 𝑦 ∈ {𝑥} → 𝑥𝑦)
1611, 15simplbiim 659 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑦)
1716adantl 481 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑦)
1817ad2antlr 763 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑦)
19 simpr 476 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20 simpl 472 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
2120adantr 480 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22 eqid 2651 . . . . . . 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)))
232, 3, 4, 5, 7, 10, 18, 19, 21, 22frgrncvvdeqlem10 27288 . . . . . 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))
241, 23hasheqf1od 13182 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑥)) = (#‘(𝐺 NeighbVtx 𝑦)))
25 frgrusgr 27240 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2625, 6anim12i 589 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
2726adantr 480 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
282hashnbusgrvd 26480 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑥𝑉) → (#‘(𝐺 NeighbVtx 𝑥)) = ((VtxDeg‘𝐺)‘𝑥))
2927, 28syl 17 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑥)) = ((VtxDeg‘𝐺)‘𝑥))
3025, 9anim12i 589 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
3130adantr 480 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
322hashnbusgrvd 26480 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑦𝑉) → (#‘(𝐺 NeighbVtx 𝑦)) = ((VtxDeg‘𝐺)‘𝑦))
3331, 32syl 17 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑦)) = ((VtxDeg‘𝐺)‘𝑦))
3424, 29, 333eqtr3d 2693 . . . 4 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘𝑦))
35 frgrncvvdeq.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
3635fveq1i 6230 . . . 4 (𝐷𝑥) = ((VtxDeg‘𝐺)‘𝑥)
3735fveq1i 6230 . . . 4 (𝐷𝑦) = ((VtxDeg‘𝐺)‘𝑦)
3834, 36, 373eqtr4g 2710 . . 3 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷𝑥) = (𝐷𝑦))
3938ex 449 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
4039ralrimivva 3000 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 1523  wcel 2030  wne 2823  wnel 2926  wral 2941  Vcvv 3231  cdif 3604  {csn 4210  {cpr 4212  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  #chash 13157  Vtxcvtx 25919  Edgcedg 25984  USGraphcusgr 26089   NeighbVtx cnbgr 26269  VtxDegcvtxdg 26417   FriendGraph cfrgr 27236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-xadd 11985  df-fz 12365  df-hash 13158  df-edg 25985  df-uhgr 25998  df-ushgr 25999  df-upgr 26022  df-umgr 26023  df-uspgr 26090  df-usgr 26091  df-nbgr 26270  df-vtxdg 26418  df-frgr 27237
This theorem is referenced by:  frgrwopreglem4a  27290
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