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Theorem frgrncvvdeq 41478
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 6551 . . . . . . 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 2605 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
5 eqid 2605 . . . . . . 7 (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
6 eqid 2605 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
7 simpl 471 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑉)
87ad2antlr 758 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑉)
9 eldifi 3689 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦𝑉)
109adantl 480 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦𝑉)
1110ad2antlr 758 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦𝑉)
12 eldif 3545 . . . . . . . . . 10 (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
13 velsn 4136 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
1413biimpri 216 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 ∈ {𝑥})
1514equcoms 1932 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 ∈ {𝑥})
1615necon3bi 2803 . . . . . . . . . 10 𝑦 ∈ {𝑥} → 𝑥𝑦)
1712, 16simplbiim 656 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑦)
1817adantl 480 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑦)
1918ad2antlr 758 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑦)
20 simpr 475 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
21 simpl 471 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
2221adantr 479 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
23 eqid 2605 . . . . . . 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 41477 . . . . . 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 12954 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (#‘(𝐺 NeighbVtx 𝑥)) = (#‘(𝐺 NeighbVtx 𝑦)))
26 frgrusgr 41430 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
2726, 7anim12i 587 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
2827adantr 479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
293hashnbusgrvd 40742 . . . . . 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 587 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
3231adantr 479 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
333hashnbusgrvd 40742 . . . . . 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 2647 . . . 4 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘𝑦))
36 frgrncvvdeq.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
3736fveq1i 6085 . . . 4 (𝐷𝑥) = ((VtxDeg‘𝐺)‘𝑥)
3836fveq1i 6085 . . . 4 (𝐷𝑦) = ((VtxDeg‘𝐺)‘𝑦)
3935, 37, 383eqtr4g 2664 . . 3 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷𝑥) = (𝐷𝑦))
4039ex 448 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
4140ralrimivva 2949 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 382   = wceq 1474  wcel 1975  wne 2775  wnel 2776  wral 2891  Vcvv 3168  cdif 3532  {csn 4120  {cpr 4122  cmpt 4633  cfv 5786  crio 6484  (class class class)co 6523  #chash 12930  Vtxcvtx 40227  Edgcedga 40349   USGraph cusgr 40377   NeighbVtx cnbgr 40548  VtxDegcvtxdg 40679   FriendGraph cfrgr 41426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-2o 7421  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-cda 8846  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-xadd 11775  df-fz 12149  df-hash 12931  df-xnn0 40196  df-uhgr 40278  df-ushgr 40279  df-upgr 40306  df-umgr 40307  df-edga 40350  df-uspgr 40378  df-usgr 40379  df-nbgr 40552  df-vtxdg 40680  df-frgr 41427
This theorem is referenced by:  frgrwopreglem4  41482
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