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Theorem frcond3 27975
Description: The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhood of a vertex and the neighborhood of a second, different vertex have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 30-Dec-2021.)
Hypotheses
Ref Expression
frcond1.v 𝑉 = (Vtx‘𝐺)
frcond1.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frcond3 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐸   𝑥,𝐺   𝑥,𝑉

Proof of Theorem frcond3
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 frcond1.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 frcond1.e . . . . 5 𝐸 = (Edg‘𝐺)
31, 2frcond1 27972 . . . 4 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
43imp 407 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5 ssrab2 4053 . . . . . . . . . 10 {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6 sseq1 3989 . . . . . . . . . 10 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
75, 6mpbii 234 . . . . . . . . 9 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
8 vex 3495 . . . . . . . . . 10 𝑥 ∈ V
98snss 4710 . . . . . . . . 9 (𝑥𝑉 ↔ {𝑥} ⊆ 𝑉)
107, 9sylibr 235 . . . . . . . 8 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → 𝑥𝑉)
1110adantl 482 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → 𝑥𝑉)
12 frgrusgr 27967 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
131, 2nbusgr 27058 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
141, 2nbusgr 27058 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
1513, 14ineq12d 4187 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
1612, 15syl 17 . . . . . . . . . . 11 (𝐺 ∈ FriendGraph → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
1716adantr 481 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
1817adantr 481 . . . . . . . . 9 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
19 inrab 4272 . . . . . . . . 9 ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
2018, 19syl6eq 2869 . . . . . . . 8 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)})
21 prcom 4660 . . . . . . . . . . . . . 14 {𝐶, 𝑏} = {𝑏, 𝐶}
2221eleq1i 2900 . . . . . . . . . . . . 13 ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
2322anbi2i 622 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24 prex 5323 . . . . . . . . . . . . 13 {𝐴, 𝑏} ∈ V
25 prex 5323 . . . . . . . . . . . . 13 {𝑏, 𝐶} ∈ V
2624, 25prss 4745 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2723, 26bitri 276 . . . . . . . . . . 11 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2827a1i 11 . . . . . . . . . 10 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ 𝑏𝑉) → (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
2928rabbidva 3476 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
3029adantr 481 . . . . . . . 8 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
31 simpr 485 . . . . . . . 8 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
3220, 30, 313eqtrd 2857 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
3311, 32jca 512 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3433ex 413 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
3534eximdv 1909 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → (∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36 reusn 4655 . . . 4 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37 df-rex 3141 . . . 4 (∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥} ↔ ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3835, 36, 373imtr4g 297 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
394, 38mpd 15 . 2 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
4039ex 413 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wrex 3136  ∃!wreu 3137  {crab 3139  cin 3932  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  Vtxcvtx 26708  Edgcedg 26759  USGraphcusgr 26861   NeighbVtx cnbgr 27041   FriendGraph cfrgr 27964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-hash 13679  df-edg 26760  df-upgr 26794  df-umgr 26795  df-usgr 26863  df-nbgr 27042  df-frgr 27965
This theorem is referenced by:  frcond4  27976  frgrncvvdeqlem3  28007
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