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Theorem frisusgranb 25124
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 25118 . 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 3581 . . . . . . . . . . . 12  |-  { n  e.  V  |  { { n ,  k } ,  { n ,  l } }  C_ 
ran  E }  C_  V
3 sseq1 3520 . . . . . . . . . . . 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 3112 . . . . . . . . . . . 12  |-  x  e. 
_V
65snss 4156 . . . . . . . . . . 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 24555 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  k )  =  {
n  e.  V  |  { k ,  n }  e.  ran  E }
)
10 nbusgra 24555 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  l )  =  {
n  e.  V  |  { l ,  n }  e.  ran  E }
)
119, 10ineq12d 3697 . . . . . . . . . . . 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 3777 . . . . . . . . . . 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 2514 . . . . . . . . . 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 4110 . . . . . . . . . . . . . . . 16  |-  { k ,  n }  =  { n ,  k }
1615eleq1i 2534 . . . . . . . . . . . . . . 15  |-  ( { k ,  n }  e.  ran  E  <->  { n ,  k }  e.  ran  E )
17 prcom 4110 . . . . . . . . . . . . . . . 16  |-  { l ,  n }  =  { n ,  l }
1817eleq1i 2534 . . . . . . . . . . . . . . 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 4696 . . . . . . . . . . . . . . 15  |-  { n ,  k }  e.  _V
21 zfpair2 4696 . . . . . . . . . . . . . . 15  |-  { n ,  l }  e.  _V
2220, 21prss 4186 . . . . . . . . . . . . . 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 3100 . . . . . . . . . . 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 2502 . . . . . . . . 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 1711 . . . . . 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 4105 . . . . . 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 2813 . . . . . 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 2865 . . . 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 2865 . . 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 1395   E.wex 1613    e. wcel 1819   A.wral 2807   E.wrex 2808   E!wreu 2809   {crab 2811    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   {cpr 4034   <.cop 4038   class class class wbr 4456   ran crn 5009  (class class class)co 6296   USGrph cusg 24457   Neighbors cnbgra 24544   FriendGrph cfrgra 25115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-hash 12409  df-usgra 24460  df-nbgra 24547  df-frgra 25116
This theorem is referenced by:  frgrancvvdeqlem4  25160
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