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Theorem frgrncvvdeq 27546
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 6875 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 NeighbVtx 𝑥) ∈ V)
2 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
3 eqid 2764 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
4 eqid 2764 . . . . . . 7 (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5 eqid 2764 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6 simpl 474 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑉)
76ad2antlr 718 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑉)
8 eldifi 3893 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦𝑉)
98adantl 473 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦𝑉)
109ad2antlr 718 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦𝑉)
11 eldif 3741 . . . . . . . . . 10 (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12 velsn 4349 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
1312biimpri 219 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 ∈ {𝑥})
1413equcoms 2117 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 ∈ {𝑥})
1514necon3bi 2962 . . . . . . . . . 10 𝑦 ∈ {𝑥} → 𝑥𝑦)
1611, 15simplbiim 499 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑦)
1716adantl 473 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑦)
1817ad2antlr 718 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑦)
19 simpr 477 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20 simpl 474 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
2120adantr 472 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22 eqid 2764 . . . . . . 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 27545 . . . . . 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 13345 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25 frgrusgr 27497 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2625, 6anim12i 606 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
2726adantr 472 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
282hashnbusgrvd 26714 . . . . . 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 606 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
3130adantr 472 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
322hashnbusgrvd 26714 . . . . . 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 2806 . . . 4 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘𝑦))
35 frgrncvvdeq.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
3635fveq1i 6375 . . . 4 (𝐷𝑥) = ((VtxDeg‘𝐺)‘𝑥)
3735fveq1i 6375 . . . 4 (𝐷𝑦) = ((VtxDeg‘𝐺)‘𝑦)
3834, 36, 373eqtr4g 2823 . . 3 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷𝑥) = (𝐷𝑦))
3938ex 401 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
4039ralrimivva 3117 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 384   = wceq 1652  wcel 2155  wne 2936  wnel 3039  wral 3054  Vcvv 3349  cdif 3728  {csn 4333  {cpr 4335  cmpt 4887  cfv 6067  crio 6801  (class class class)co 6841  chash 13320  Vtxcvtx 26164  Edgcedg 26215  USGraphcusgr 26321   NeighbVtx cnbgr 26502  VtxDegcvtxdg 26651   FriendGraph cfrgr 27493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-2o 7764  df-oadd 7767  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-card 9015  df-cda 9242  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-2 11334  df-n0 11538  df-xnn0 11610  df-z 11624  df-uz 11886  df-xadd 12146  df-fz 12533  df-hash 13321  df-edg 26216  df-uhgr 26229  df-ushgr 26230  df-upgr 26253  df-umgr 26254  df-uspgr 26322  df-usgr 26323  df-nbgr 26503  df-vtxdg 26652  df-frgr 27494
This theorem is referenced by:  frgrwopreglem4a  27547
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