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Theorem frcond3 41440
 Description: The friendship condition, expressed by neighborhoods: in a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 29-Mar-2021.)
Hypotheses
Ref Expression
frcond1.v 𝑉 = (Vtx‘𝐺)
frcond1.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frcond3 (𝐺 ∈ FriendGraph → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})
Distinct variable groups:   𝑘,𝑙,𝐸   𝑘,𝐺,𝑙   𝑘,𝑉,𝑙   𝑥,𝐸,𝑘,𝑙   𝑥,𝐺   𝑥,𝑉

Proof of Theorem frcond3
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 frcond1.v . . 3 𝑉 = (Vtx‘𝐺)
2 frcond1.e . . 3 𝐸 = (Edg‘𝐺)
31, 2frgrusgrfrcond 41431 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸))
4 ssrab2 3650 . . . . . . . . . . 11 {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} ⊆ 𝑉
5 sseq1 3589 . . . . . . . . . . 11 ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
64, 5mpbii 222 . . . . . . . . . 10 ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
7 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
87snss 4259 . . . . . . . . . 10 (𝑥𝑉 ↔ {𝑥} ⊆ 𝑉)
96, 8sylibr 223 . . . . . . . . 9 ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → 𝑥𝑉)
109adantl 481 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → 𝑥𝑉)
111, 2nbusgr 40571 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑘) = {𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸})
121, 2nbusgr 40571 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑙) = {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸})
1311, 12ineq12d 3777 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}))
1413adantr 480 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}))
1514adantr 480 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}))
16 inrab 3858 . . . . . . . . . 10 ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}) = {𝑛𝑉 ∣ ({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸)}
1715, 16syl6eq 2660 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑛𝑉 ∣ ({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸)})
18 prcom 4211 . . . . . . . . . . . . . . 15 {𝑘, 𝑛} = {𝑛, 𝑘}
1918eleq1i 2679 . . . . . . . . . . . . . 14 ({𝑘, 𝑛} ∈ 𝐸 ↔ {𝑛, 𝑘} ∈ 𝐸)
20 prcom 4211 . . . . . . . . . . . . . . 15 {𝑙, 𝑛} = {𝑛, 𝑙}
2120eleq1i 2679 . . . . . . . . . . . . . 14 ({𝑙, 𝑛} ∈ 𝐸 ↔ {𝑛, 𝑙} ∈ 𝐸)
2219, 21anbi12i 729 . . . . . . . . . . . . 13 (({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸) ↔ ({𝑛, 𝑘} ∈ 𝐸 ∧ {𝑛, 𝑙} ∈ 𝐸))
23 zfpair2 4834 . . . . . . . . . . . . . 14 {𝑛, 𝑘} ∈ V
24 zfpair2 4834 . . . . . . . . . . . . . 14 {𝑛, 𝑙} ∈ V
2523, 24prss 4291 . . . . . . . . . . . . 13 (({𝑛, 𝑘} ∈ 𝐸 ∧ {𝑛, 𝑙} ∈ 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸)
2622, 25bitri 263 . . . . . . . . . . . 12 (({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸)
2726a1i 11 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ 𝑛𝑉) → (({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸))
2827rabbidva 3163 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → {𝑛𝑉 ∣ ({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸)} = {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸})
2928adantr 480 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → {𝑛𝑉 ∣ ({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸)} = {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸})
30 simpr 476 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥})
3117, 29, 303eqtrd 2648 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})
3210, 31jca 553 . . . . . . 7 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3332ex 449 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})))
3433eximdv 1833 . . . . 5 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → (∃𝑥{𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})))
35 reusn 4206 . . . . 5 (∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸 ↔ ∃𝑥{𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥})
36 df-rex 2902 . . . . 5 (∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥} ↔ ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3734, 35, 363imtr4g 284 . . . 4 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → (∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸 → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3837ralimdvva 2947 . . 3 (𝐺 ∈ USGraph → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸 → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3938imp 444 . 2 ((𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸) → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})
403, 39sylbi 206 1 (𝐺 ∈ FriendGraph → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-umgr 25750  df-edga 25793  df-usgr 40381  df-nbgr 40554  df-frgr 41429 This theorem is referenced by:  frgrncvvdeqlem4  41472
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