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Theorem frcond3 27964
 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 27961 . . . 4 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
43imp 407 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5 ssrab2 4059 . . . . . . . . . 10 {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6 sseq1 3995 . . . . . . . . . 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 3502 . . . . . . . . . 10 𝑥 ∈ V
98snss 4716 . . . . . . . . 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 27956 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
131, 2nbusgr 27047 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
141, 2nbusgr 27047 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
1513, 14ineq12d 4193 . . . . . . . . . . . 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 4278 . . . . . . . . 9 ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
2018, 19syl6eq 2876 . . . . . . . 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 4666 . . . . . . . . . . . . . 14 {𝐶, 𝑏} = {𝑏, 𝐶}
2221eleq1i 2907 . . . . . . . . . . . . 13 ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
2322anbi2i 622 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24 prex 5328 . . . . . . . . . . . . 13 {𝐴, 𝑏} ∈ V
25 prex 5328 . . . . . . . . . . . . 13 {𝑏, 𝐶} ∈ V
2624, 25prss 4751 . . . . . . . . . . . 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 3483 . . . . . . . . 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 2864 . . . . . . 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 1911 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → (∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36 reusn 4661 . . . 4 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37 df-rex 3148 . . . 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 1081   = wceq 1530  ∃wex 1773   ∈ wcel 2107   ≠ wne 3020  ∃wrex 3143  ∃!wreu 3144  {crab 3146   ∩ cin 3938   ⊆ wss 3939  {csn 4563  {cpr 4565  ‘cfv 6351  (class class class)co 7151  Vtxcvtx 26697  Edgcedg 26748  USGraphcusgr 26850   NeighbVtx cnbgr 27030   FriendGraph cfrgr 27953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-edg 26749  df-upgr 26783  df-umgr 26784  df-usgr 26852  df-nbgr 27031  df-frgr 27954 This theorem is referenced by:  frcond4  27965  frgrncvvdeqlem3  27996
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