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Theorem frcond3 27610
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 27607 . . . 4 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
43imp 396 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5 ssrab2 3881 . . . . . . . . . 10 {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6 sseq1 3820 . . . . . . . . . 10 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
75, 6mpbii 225 . . . . . . . . 9 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
8 vex 3386 . . . . . . . . . 10 𝑥 ∈ V
98snss 4502 . . . . . . . . 9 (𝑥𝑉 ↔ {𝑥} ⊆ 𝑉)
107, 9sylibr 226 . . . . . . . 8 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → 𝑥𝑉)
1110adantl 474 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → 𝑥𝑉)
12 frgrusgr 27601 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
131, 2nbusgr 26579 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
141, 2nbusgr 26579 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
1513, 14ineq12d 4011 . . . . . . . . . . . 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 473 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
1817adantr 473 . . . . . . . . 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 4097 . . . . . . . . 9 ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
2018, 19syl6eq 2847 . . . . . . . 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 4454 . . . . . . . . . . . . . 14 {𝐶, 𝑏} = {𝑏, 𝐶}
2221eleq1i 2867 . . . . . . . . . . . . 13 ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
2322anbi2i 617 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24 prex 5098 . . . . . . . . . . . . 13 {𝐴, 𝑏} ∈ V
25 prex 5098 . . . . . . . . . . . . 13 {𝑏, 𝐶} ∈ V
2624, 25prss 4537 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2723, 26bitri 267 . . . . . . . . . . 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 3370 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
3029adantr 473 . . . . . . . 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 478 . . . . . . . 8 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
3220, 30, 313eqtrd 2835 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
3311, 32jca 508 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3433ex 402 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
3534eximdv 2013 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → (∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36 reusn 4449 . . . 4 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37 df-rex 3093 . . . 4 (∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥} ↔ ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3835, 36, 373imtr4g 288 . . 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 402 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  wne 2969  wrex 3088  ∃!wreu 3089  {crab 3091  cin 3766  wss 3767  {csn 4366  {cpr 4368  cfv 6099  (class class class)co 6876  Vtxcvtx 26223  Edgcedg 26274  USGraphcusgr 26377   NeighbVtx cnbgr 26558   FriendGraph cfrgr 27597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-2o 7798  df-oadd 7801  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-card 9049  df-cda 9276  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-n0 11577  df-xnn0 11649  df-z 11663  df-uz 11927  df-fz 12577  df-hash 13367  df-edg 26275  df-upgr 26309  df-umgr 26310  df-usgr 26379  df-nbgr 26559  df-frgr 27598
This theorem is referenced by:  frcond4  27611  frgrncvvdeqlem3  27642
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