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Theorem frcond3 29216
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 29213 . . . 4 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸))
43imp 408 . . 3 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
5 ssrab2 4038 . . . . . . . . . 10 {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉
6 sseq1 3970 . . . . . . . . . 10 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} ⊆ 𝑉 ↔ {𝑥} ⊆ 𝑉))
75, 6mpbii 232 . . . . . . . . 9 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → {𝑥} ⊆ 𝑉)
8 vex 3450 . . . . . . . . . 10 𝑥 ∈ V
98snss 4747 . . . . . . . . 9 (𝑥𝑉 ↔ {𝑥} ⊆ 𝑉)
107, 9sylibr 233 . . . . . . . 8 ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → 𝑥𝑉)
1110adantl 483 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → 𝑥𝑉)
12 frgrusgr 29208 . . . . . . . . . . . 12 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
131, 2nbusgr 28300 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐴) = {𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸})
141, 2nbusgr 28300 . . . . . . . . . . . . 13 (𝐺 ∈ USGraph → (𝐺 NeighbVtx 𝐶) = {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸})
1513, 14ineq12d 4174 . . . . . . . . . . . 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 482 . . . . . . . . . 10 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}))
1817adantr 482 . . . . . . . . 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 4267 . . . . . . . . 9 ({𝑏𝑉 ∣ {𝐴, 𝑏} ∈ 𝐸} ∩ {𝑏𝑉 ∣ {𝐶, 𝑏} ∈ 𝐸}) = {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)}
2018, 19eqtrdi 2793 . . . . . . . 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 4694 . . . . . . . . . . . . . 14 {𝐶, 𝑏} = {𝑏, 𝐶}
2221eleq1i 2829 . . . . . . . . . . . . 13 ({𝐶, 𝑏} ∈ 𝐸 ↔ {𝑏, 𝐶} ∈ 𝐸)
2322anbi2i 624 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸) ↔ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))
24 prex 5390 . . . . . . . . . . . . 13 {𝐴, 𝑏} ∈ V
25 prex 5390 . . . . . . . . . . . . 13 {𝑏, 𝐶} ∈ V
2624, 25prss 4781 . . . . . . . . . . . 12 (({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸) ↔ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸)
2723, 26bitri 275 . . . . . . . . . . 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 3415 . . . . . . . . 9 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → {𝑏𝑉 ∣ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝐶, 𝑏} ∈ 𝐸)} = {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸})
3029adantr 482 . . . . . . . 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 486 . . . . . . . 8 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
3220, 30, 313eqtrd 2781 . . . . . . 7 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})
3311, 32jca 513 . . . . . 6 (((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) ∧ {𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥}) → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3433ex 414 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → ({𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → (𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
3534eximdv 1921 . . . 4 ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐶𝑉𝐴𝐶)) → (∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥} → ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥})))
36 reusn 4689 . . . 4 (∃!𝑏𝑉 {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸 ↔ ∃𝑥{𝑏𝑉 ∣ {{𝐴, 𝑏}, {𝑏, 𝐶}} ⊆ 𝐸} = {𝑥})
37 df-rex 3075 . . . 4 (∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥} ↔ ∃𝑥(𝑥𝑉 ∧ ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
3835, 36, 373imtr4g 296 . . 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 414 1 (𝐺 ∈ FriendGraph → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃𝑥𝑉 ((𝐺 NeighbVtx 𝐴) ∩ (𝐺 NeighbVtx 𝐶)) = {𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wne 2944  wrex 3074  ∃!wreu 3352  {crab 3408  cin 3910  wss 3911  {csn 4587  {cpr 4589  cfv 6497  (class class class)co 7358  Vtxcvtx 27950  Edgcedg 28001  USGraphcusgr 28103   NeighbVtx cnbgr 28283   FriendGraph cfrgr 29205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-n0 12415  df-xnn0 12487  df-z 12501  df-uz 12765  df-fz 13426  df-hash 14232  df-edg 28002  df-upgr 28036  df-umgr 28037  df-usgr 28105  df-nbgr 28284  df-frgr 29206
This theorem is referenced by:  frcond4  29217  frgrncvvdeqlem3  29248
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