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Theorem frcond3 28158
 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 28155 . . . 4 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
43imp 410 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5 ssrab2 3986 . . . . . . . . . 10 {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6 sseq1 3919 . . . . . . . . . 10 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
75, 6mpbii 236 . . . . . . . . 9 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
8 vex 3413 . . . . . . . . . 10 𝑥 ∈ V
98snss 4679 . . . . . . . . 9 (𝑥𝑉 ↔ {𝑥} ⊆ 𝑉)
107, 9sylibr 237 . . . . . . . 8 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → 𝑥𝑉)
1110adantl 485 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → 𝑥𝑉)
12 frgrusgr 28150 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
131, 2nbusgr 27243 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
141, 2nbusgr 27243 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
1513, 14ineq12d 4120 . . . . . . . . . . . 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 484 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
1817adantr 484 . . . . . . . . 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 4211 . . . . . . . . 9 ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
2018, 19eqtrdi 2809 . . . . . . . 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 4628 . . . . . . . . . . . . . 14 {𝐶, 𝑏} = {𝑏, 𝐶}
2221eleq1i 2842 . . . . . . . . . . . . 13 ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
2322anbi2i 625 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24 prex 5304 . . . . . . . . . . . . 13 {𝐴, 𝑏} ∈ V
25 prex 5304 . . . . . . . . . . . . 13 {𝑏, 𝐶} ∈ V
2624, 25prss 4713 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2723, 26bitri 278 . . . . . . . . . . 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 3390 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
3029adantr 484 . . . . . . . 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 488 . . . . . . . 8 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
3220, 30, 313eqtrd 2797 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
3311, 32jca 515 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3433ex 416 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
3534eximdv 1918 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → (∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36 reusn 4623 . . . 4 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37 df-rex 3076 . . . 4 (∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥} ↔ ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3835, 36, 373imtr4g 299 . . 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 416 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  ∃!wreu 3072  {crab 3074   ∩ cin 3859   ⊆ wss 3860  {csn 4525  {cpr 4527  ‘cfv 6339  (class class class)co 7155  Vtxcvtx 26893  Edgcedg 26944  USGraphcusgr 27046   NeighbVtx cnbgr 27226   FriendGraph cfrgr 28147 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 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 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-oadd 8121  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-dju 9368  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-n0 11940  df-xnn0 12012  df-z 12026  df-uz 12288  df-fz 12945  df-hash 13746  df-edg 26945  df-upgr 26979  df-umgr 26980  df-usgr 27048  df-nbgr 27227  df-frgr 28148 This theorem is referenced by:  frcond4  28159  frgrncvvdeqlem3  28190
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