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Theorem frisusgranb 30760
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 30754 . 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 3548 . . . . . . . . . . . 12  |-  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  C_  V
3 sseq1 3488 . . . . . . . . . . . 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 211 . . . . . . . . . . 11  |-  ( { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x }  ->  { x }  C_  V
)
5 vex 3081 . . . . . . . . . . . 12  |-  x  e. 
_V
65snss 4110 . . . . . . . . . . 11  |-  ( x  e.  V  <->  { x }  C_  V )
74, 6sylibr 212 . . . . . . . . . 10  |-  ( { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  =  { x }  ->  x  e.  V )
87adantl 466 . . . . . . . . 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 23518 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  k )  =  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)
10 nbusgra 23518 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  l )  =  {
n  e.  V  |  { l ,  n }  e.  ran  E }
)
119, 10ineq12d 3664 . . . . . . . . . . . 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 729 . . . . . . . . . . 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 3733 . . . . . . . . . . 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 2511 . . . . . . . . . 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 4064 . . . . . . . . . . . . . . . 16  |-  { k ,  n }  =  { n ,  k }
1615eleq1i 2531 . . . . . . . . . . . . . . 15  |-  ( { k ,  n }  e.  ran  E  <->  { n ,  k }  e.  ran  E )
17 prcom 4064 . . . . . . . . . . . . . . . 16  |-  { l ,  n }  =  { n ,  l }
1817eleq1i 2531 . . . . . . . . . . . . . . 15  |-  ( { l ,  n }  e.  ran  E  <->  { n ,  l }  e.  ran  E )
1916, 18anbi12i 697 . . . . . . . . . . . . . 14  |-  ( ( { k ,  n }  e.  ran  E  /\  { l ,  n }  e.  ran  E )  <->  ( {
n ,  k }  e.  ran  E  /\  { n ,  l }  e.  ran  E ) )
20 zfpair2 4643 . . . . . . . . . . . . . . 15  |-  { n ,  k }  e.  _V
21 zfpair2 4643 . . . . . . . . . . . . . . 15  |-  { n ,  l }  e.  _V
2220, 21prss 4138 . . . . . . . . . . . . . 14  |-  ( ( { n ,  k }  e.  ran  E  /\  { n ,  l }  e.  ran  E
)  <->  { { n ,  k } ,  {
n ,  l } }  C_  ran  E )
2319, 22bitri 249 . . . . . . . . . . . . 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 3069 . . . . . . . . . . 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 465 . . . . . . . . . 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 461 . . . . . . . . . 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 2499 . . . . . . . . 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 532 . . . . . . . 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 434 . . . . . . 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 1677 . . . . . 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 4059 . . . . . 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 2805 . . . . . 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 270 . . . . 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 2832 . . . 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 2832 . . 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 429 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   A.wral 2799   E.wrex 2800   E!wreu 2801   {crab 2803    \ cdif 3436    i^i cin 3438    C_ wss 3439   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403   ran crn 4952  (class class class)co 6203   USGrph cusg 23443   Neighbors cnbgra 23508   FriendGrph cfrgra 30751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-hash 12225  df-usgra 23445  df-nbgra 23511  df-frgra 30752
This theorem is referenced by:  frgrancvvdeqlem4  30797
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