Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frisusgranb Unicode version

Theorem frisusgranb 28109
Description: 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.)
Assertion
Ref Expression
frisusgranb  |-  ( V FriendGrph  E  ->  A. k  e.  V  A. l  e.  ( V  \  { k } ) E. x  e.  V  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } )
Distinct variable groups:    k, V, l, x    k, E, l, x

Proof of Theorem frisusgranb
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 frisusgrapr 28103 . 2  |-  ( V FriendGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V  \  { k } ) E! n  e.  V  { { n ,  k } ,  { n ,  l } }  C_  ran  E ) )
2 ssrab2 3396 . . . . . . . . . . . 12  |-  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  C_  V
3 sseq1 3337 . . . . . . . . . . . 12  |-  ( { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x }  ->  ( { n  e.  V  |  { { n ,  k } ,  {
n ,  l } }  C_  ran  E }  C_  V  <->  { x }  C_  V ) )
42, 3mpbii 203 . . . . . . . . . . 11  |-  ( { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x }  ->  { x }  C_  V
)
5 vex 2927 . . . . . . . . . . . 12  |-  x  e. 
_V
65snss 3894 . . . . . . . . . . 11  |-  ( x  e.  V  <->  { x }  C_  V )
74, 6sylibr 204 . . . . . . . . . 10  |-  ( { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x }  ->  x  e.  V )
87adantl 453 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  x  e.  V
)
9 nbusgra 21401 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  k )  =  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)
10 nbusgra 21401 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  l )  =  {
n  e.  V  |  { l ,  n }  e.  ran  E }
)
119, 10ineq12d 3511 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  ( { n  e.  V  |  { k ,  n }  e.  ran  E }  i^i  { n  e.  V  |  { l ,  n }  e.  ran  E }
) )
1211ad3antrrr 711 . . . . . . . . . . 11  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  ( { n  e.  V  |  { k ,  n }  e.  ran  E }  i^i  { n  e.  V  |  { l ,  n }  e.  ran  E }
) )
13 inrab 3581 . . . . . . . . . . 11  |-  ( { n  e.  V  |  { k ,  n }  e.  ran  E }  i^i  { n  e.  V  |  { l ,  n }  e.  ran  E }
)  =  { n  e.  V  |  ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E ) }
1412, 13syl6eq 2460 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { n  e.  V  |  ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E ) } )
15 prcom 3850 . . . . . . . . . . . . . . . 16  |-  { k ,  n }  =  { n ,  k }
1615eleq1i 2475 . . . . . . . . . . . . . . 15  |-  ( { k ,  n }  e.  ran  E  <->  { n ,  k }  e.  ran  E )
17 prcom 3850 . . . . . . . . . . . . . . . 16  |-  { l ,  n }  =  { n ,  l }
1817eleq1i 2475 . . . . . . . . . . . . . . 15  |-  ( { l ,  n }  e.  ran  E  <->  { n ,  l }  e.  ran  E )
1916, 18anbi12i 679 . . . . . . . . . . . . . 14  |-  ( ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E )  <->  ( {
n ,  k }  e.  ran  E  /\  { n ,  l }  e.  ran  E ) )
20 zfpair2 4372 . . . . . . . . . . . . . . 15  |-  { n ,  k }  e.  _V
21 zfpair2 4372 . . . . . . . . . . . . . . 15  |-  { n ,  l }  e.  _V
2220, 21prss 3920 . . . . . . . . . . . . . 14  |-  ( ( { n ,  k }  e.  ran  E  /\  { n ,  l }  e.  ran  E
)  <->  { { n ,  k } ,  {
n ,  l } }  C_  ran  E )
2319, 22bitri 241 . . . . . . . . . . . . 13  |-  ( ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E )  <->  { { n ,  k } ,  { n ,  l } }  C_  ran  E )
2423a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  n  e.  V )  ->  ( ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E )  <->  { { n ,  k } ,  { n ,  l } }  C_  ran  E ) )
2524rabbidva 2915 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  k  e.  V )  /\  l  e.  ( V  \  { k } ) )  ->  { n  e.  V  |  ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E ) }  =  { n  e.  V  |  { {
n ,  k } ,  { n ,  l } }  C_  ran  E } )
2625adantr 452 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  { n  e.  V  |  ( { k ,  n }  e.  ran  E  /\  {
l ,  n }  e.  ran  E ) }  =  { n  e.  V  |  { {
n ,  k } ,  { n ,  l } }  C_  ran  E } )
27 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  { n  e.  V  |  { {
n ,  k } ,  { n ,  l } }  C_  ran  E }  =  {
x } )
2814, 26, 273eqtrd 2448 . . . . . . . . 9  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } )
298, 28jca 519 . . . . . . . 8  |-  ( ( ( ( V USGrph  E  /\  k  e.  V
)  /\  l  e.  ( V  \  { k } ) )  /\  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )  ->  ( x  e.  V  /\  ( (
<. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } ) )
3029ex 424 . . . . . . 7  |-  ( ( ( V USGrph  E  /\  k  e.  V )  /\  l  e.  ( V  \  { k } ) )  ->  ( { n  e.  V  |  { { n ,  k } ,  {
n ,  l } }  C_  ran  E }  =  { x }  ->  ( x  e.  V  /\  ( ( <. V ,  E >. Neighbors  k )  i^i  ( <. V ,  E >. Neighbors  l
) )  =  {
x } ) ) )
3130eximdv 1629 . . . . . 6  |-  ( ( ( V USGrph  E  /\  k  e.  V )  /\  l  e.  ( V  \  { k } ) )  ->  ( E. x { n  e.  V  |  { {
n ,  k } ,  { n ,  l } }  C_  ran  E }  =  {
x }  ->  E. x
( x  e.  V  /\  ( ( <. V ,  E >. Neighbors  k )  i^i  ( <. V ,  E >. Neighbors  l
) )  =  {
x } ) ) )
32 reusn 3845 . . . . . 6  |-  ( E! n  e.  V  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E  <->  E. x { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x } )
33 df-rex 2680 . . . . . 6  |-  ( E. x  e.  V  ( ( <. V ,  E >. Neighbors 
k )  i^i  ( <. V ,  E >. Neighbors  l
) )  =  {
x }  <->  E. x
( x  e.  V  /\  ( ( <. V ,  E >. Neighbors  k )  i^i  ( <. V ,  E >. Neighbors  l
) )  =  {
x } ) )
3431, 32, 333imtr4g 262 . . . . 5  |-  ( ( ( V USGrph  E  /\  k  e.  V )  /\  l  e.  ( V  \  { k } ) )  ->  ( E! n  e.  V  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E  ->  E. x  e.  V  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } ) )
3534ralimdva 2752 . . . 4  |-  ( ( V USGrph  E  /\  k  e.  V )  ->  ( A. l  e.  ( V  \  { k } ) E! n  e.  V  { { n ,  k } ,  { n ,  l } }  C_  ran  E  ->  A. l  e.  ( V  \  { k } ) E. x  e.  V  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } ) )
3635ralimdva 2752 . . 3  |-  ( V USGrph  E  ->  ( A. k  e.  V  A. l  e.  ( V  \  {
k } ) E! n  e.  V  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E  ->  A. k  e.  V  A. l  e.  ( V  \  {
k } ) E. x  e.  V  ( ( <. V ,  E >. Neighbors 
k )  i^i  ( <. V ,  E >. Neighbors  l
) )  =  {
x } ) )
3736imp 419 . 2  |-  ( ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V  \  {
k } ) E! n  e.  V  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E )  ->  A. k  e.  V  A. l  e.  ( V  \  { k } ) E. x  e.  V  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } )
381, 37syl 16 1  |-  ( V FriendGrph  E  ->  A. k  e.  V  A. l  e.  ( V  \  { k } ) E. x  e.  V  ( ( <. V ,  E >. Neighbors  k
)  i^i  ( <. V ,  E >. Neighbors  l ) )  =  { x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675   E!wreu 2676   {crab 2678    \ cdif 3285    i^i cin 3287    C_ wss 3288   {csn 3782   {cpr 3783   <.cop 3785   class class class wbr 4180   ran crn 4846  (class class class)co 6048   USGrph cusg 21326   Neighbors cnbgra 21391   FriendGrph cfrgra 28100
This theorem is referenced by:  frgrancvvdeqlem4  28144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-hash 11582  df-usgra 21328  df-nbgra 21394  df-frgra 28101
  Copyright terms: Public domain W3C validator