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Theorem frcond3 41435
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 41426 . 2 (𝐺 ∈ FriendGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸))
4 ssrab2 3649 . . . . . . . . . . 11 {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} ⊆ 𝑉
5 sseq1 3588 . . . . . . . . . . 11 ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
64, 5mpbii 221 . . . . . . . . . 10 ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
7 vex 3175 . . . . . . . . . . 11 𝑥 ∈ V
87snss 4258 . . . . . . . . . 10 (𝑥𝑉 ↔ {𝑥} ⊆ 𝑉)
96, 8sylibr 222 . . . . . . . . 9 ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → 𝑥𝑉)
109adantl 480 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → 𝑥𝑉)
111, 2nbusgr 40566 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑘) = {𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸})
121, 2nbusgr 40566 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝑙) = {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸})
1311, 12ineq12d 3776 . . . . . . . . . . . 12 (𝐺 ∈ USGraph → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}))
1413adantr 479 . . . . . . . . . . 11 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}))
1514adantr 479 . . . . . . . . . 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 3857 . . . . . . . . . 10 ({𝑛𝑉 ∣ {𝑘, 𝑛} ∈ 𝐸} ∩ {𝑛𝑉 ∣ {𝑙, 𝑛} ∈ 𝐸}) = {𝑛𝑉 ∣ ({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸)}
1715, 16syl6eq 2659 . . . . . . . . 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 4210 . . . . . . . . . . . . . . 15 {𝑘, 𝑛} = {𝑛, 𝑘}
1918eleq1i 2678 . . . . . . . . . . . . . 14 ({𝑘, 𝑛} ∈ 𝐸 ↔ {𝑛, 𝑘} ∈ 𝐸)
20 prcom 4210 . . . . . . . . . . . . . . 15 {𝑙, 𝑛} = {𝑛, 𝑙}
2120eleq1i 2678 . . . . . . . . . . . . . 14 ({𝑙, 𝑛} ∈ 𝐸 ↔ {𝑛, 𝑙} ∈ 𝐸)
2219, 21anbi12i 728 . . . . . . . . . . . . 13 (({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸) ↔ ({𝑛, 𝑘} ∈ 𝐸 ∧ {𝑛, 𝑙} ∈ 𝐸))
23 zfpair2 4829 . . . . . . . . . . . . . 14 {𝑛, 𝑘} ∈ V
24 zfpair2 4829 . . . . . . . . . . . . . 14 {𝑛, 𝑙} ∈ V
2523, 24prss 4290 . . . . . . . . . . . . 13 (({𝑛, 𝑘} ∈ 𝐸 ∧ {𝑛, 𝑙} ∈ 𝐸) ↔ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸)
2622, 25bitri 262 . . . . . . . . . . . 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 3162 . . . . . . . . . 10 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → {𝑛𝑉 ∣ ({𝑘, 𝑛} ∈ 𝐸 ∧ {𝑙, 𝑛} ∈ 𝐸)} = {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸})
2928adantr 479 . . . . . . . . 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 475 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥})
3117, 29, 303eqtrd 2647 . . . . . . . 8 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})
3210, 31jca 552 . . . . . . 7 (((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) ∧ {𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥}) → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3332ex 448 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → ({𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥} → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥})))
3433eximdv 1832 . . . . 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 4205 . . . . 5 (∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸 ↔ ∃𝑥{𝑛𝑉 ∣ {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸} = {𝑥})
36 df-rex 2901 . . . . 5 (∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥} ↔ ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3734, 35, 363imtr4g 283 . . . 4 ((𝐺 ∈ USGraph ∧ (𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘}))) → (∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸 → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3837ralimdvva 2946 . . 3 (𝐺 ∈ USGraph → (∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃!𝑛𝑉 {{𝑛, 𝑘}, {𝑛, 𝑙}} ⊆ 𝐸 → ∀𝑘𝑉𝑙 ∈ (𝑉 ∖ {𝑘})∃𝑥𝑉 ((𝐺 NeighbVtx 𝑘) ∩ (𝐺 NeighbVtx 𝑙)) = {𝑥}))
3938imp 443 . 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 205 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 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wral 2895  wrex 2896  ∃!wreu 2897  {crab 2899  cdif 3536  cin 3538  wss 3539  {csn 4124  {cpr 4126  cfv 5790  (class class class)co 6527  Vtxcvtx 40224  Edgcedga 40346   USGraph cusgr 40374   NeighbVtx cnbgr 40545   FriendGraph cfrgr 41423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-hash 12935  df-upgr 40303  df-umgr 40304  df-edga 40347  df-usgr 40376  df-nbgr 40549  df-frgr 41424
This theorem is referenced by:  frgrncvvdeqlem4  41467
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