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Theorem frisusgranb 28387
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 28381 . 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 3428 . . . . . . . . . . . 12  |-  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  C_  V
3 sseq1 3369 . . . . . . . . . . . 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 2959 . . . . . . . . . . . 12  |-  x  e. 
_V
65snss 3926 . . . . . . . . . . 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 21440 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  k )  =  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)
10 nbusgra 21440 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  l )  =  {
n  e.  V  |  { l ,  n }  e.  ran  E }
)
119, 10ineq12d 3543 . . . . . . . . . . . 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 3613 . . . . . . . . . . 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 2484 . . . . . . . . . 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 3882 . . . . . . . . . . . . . . . 16  |-  { k ,  n }  =  { n ,  k }
1615eleq1i 2499 . . . . . . . . . . . . . . 15  |-  ( { k ,  n }  e.  ran  E  <->  { n ,  k }  e.  ran  E )
17 prcom 3882 . . . . . . . . . . . . . . . 16  |-  { l ,  n }  =  { n ,  l }
1817eleq1i 2499 . . . . . . . . . . . . . . 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 4404 . . . . . . . . . . . . . . 15  |-  { n ,  k }  e.  _V
21 zfpair2 4404 . . . . . . . . . . . . . . 15  |-  { n ,  l }  e.  _V
2220, 21prss 3952 . . . . . . . . . . . . . 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 2947 . . . . . . . . . . 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 2472 . . . . . . . . 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 1632 . . . . . 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 3877 . . . . . 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 2711 . . . . . 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 2784 . . . 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 2784 . . 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 1550    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709    \ cdif 3317    i^i cin 3319    C_ wss 3320   {csn 3814   {cpr 3815   <.cop 3817   class class class wbr 4212   ran crn 4879  (class class class)co 6081   USGrph cusg 21365   Neighbors cnbgra 21430   FriendGrph cfrgra 28378
This theorem is referenced by:  frgrancvvdeqlem4  28422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-hash 11619  df-usgra 21367  df-nbgra 21433  df-frgra 28379
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