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Theorem frgrncvvdeq 28072
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 7165 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 NeighbVtx 𝑥) ∈ V)
2 frgrncvvdeq.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
3 eqid 2821 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
4 eqid 2821 . . . . . . 7 (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑥)
5 eqid 2821 . . . . . . 7 (𝐺 NeighbVtx 𝑦) = (𝐺 NeighbVtx 𝑦)
6 simpl 486 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑉)
76ad2antlr 726 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑉)
8 eldifi 4079 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑦𝑉)
98adantl 485 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑦𝑉)
109ad2antlr 726 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦𝑉)
11 eldif 3920 . . . . . . . . . 10 (𝑦 ∈ (𝑉 ∖ {𝑥}) ↔ (𝑦𝑉 ∧ ¬ 𝑦 ∈ {𝑥}))
12 velsn 4556 . . . . . . . . . . . . 13 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
1312biimpri 231 . . . . . . . . . . . 12 (𝑦 = 𝑥𝑦 ∈ {𝑥})
1413equcoms 2028 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 ∈ {𝑥})
1514necon3bi 3033 . . . . . . . . . 10 𝑦 ∈ {𝑥} → 𝑥𝑦)
1611, 15simplbiim 508 . . . . . . . . 9 (𝑦 ∈ (𝑉 ∖ {𝑥}) → 𝑥𝑦)
1716adantl 485 . . . . . . . 8 ((𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})) → 𝑥𝑦)
1817ad2antlr 726 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑥𝑦)
19 simpr 488 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝑦 ∉ (𝐺 NeighbVtx 𝑥))
20 simpl 486 . . . . . . . 8 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → 𝐺 ∈ FriendGraph )
2120adantr 484 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → 𝐺 ∈ FriendGraph )
22 eqid 2821 . . . . . . 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 28071 . . . . . 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 13698 . . . . 5 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (♯‘(𝐺 NeighbVtx 𝑥)) = (♯‘(𝐺 NeighbVtx 𝑦)))
25 frgrusgr 28024 . . . . . . . 8 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
2625, 6anim12i 615 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
2726adantr 484 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑥𝑉))
282hashnbusgrvd 27296 . . . . . 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 615 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
3130adantr 484 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐺 ∈ USGraph ∧ 𝑦𝑉))
322hashnbusgrvd 27296 . . . . . 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 2864 . . . 4 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → ((VtxDeg‘𝐺)‘𝑥) = ((VtxDeg‘𝐺)‘𝑦))
35 frgrncvvdeq.d . . . . 5 𝐷 = (VtxDeg‘𝐺)
3635fveq1i 6644 . . . 4 (𝐷𝑥) = ((VtxDeg‘𝐺)‘𝑥)
3735fveq1i 6644 . . . 4 (𝐷𝑦) = ((VtxDeg‘𝐺)‘𝑦)
3834, 36, 373eqtr4g 2881 . . 3 (((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) ∧ 𝑦 ∉ (𝐺 NeighbVtx 𝑥)) → (𝐷𝑥) = (𝐷𝑦))
3938ex 416 . 2 ((𝐺 ∈ FriendGraph ∧ (𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥}))) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
4039ralrimivva 3179 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 399   = wceq 1538  wcel 2115  wne 3007  wnel 3111  wral 3126  Vcvv 3471  cdif 3907  {csn 4540  {cpr 4542  cmpt 5119  cfv 6328  crio 7087  (class class class)co 7130  chash 13674  Vtxcvtx 26767  Edgcedg 26818  USGraphcusgr 26920   NeighbVtx cnbgr 27100  VtxDegcvtxdg 27233   FriendGraph cfrgr 28021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-dju 9306  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-n0 11876  df-xnn0 11946  df-z 11960  df-uz 12222  df-xadd 12486  df-fz 12876  df-hash 13675  df-edg 26819  df-uhgr 26829  df-ushgr 26830  df-upgr 26853  df-umgr 26854  df-uspgr 26921  df-usgr 26922  df-nbgr 27101  df-vtxdg 27234  df-frgr 28022
This theorem is referenced by:  frgrwopreglem4a  28073
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