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Theorem frgra1v 27753
Description: Any graph with only one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
frgra1v  |-  ( ( V  e.  X  /\  { V } USGrph  E )  ->  { V } FriendGrph  E )

Proof of Theorem frgra1v
Dummy variables  k 
l  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 21240 . . 3  |-  ( { V } USGrph  E  ->  ( { V }  e.  _V  /\  E  e.  _V ) )
2 simplr 732 . . . . 5  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } USGrph  E )
3 ral0 3677 . . . . . 6  |-  A. l  e.  (/)  E! x  e. 
{ V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E
4 sneq 3770 . . . . . . . . . . 11  |-  ( k  =  V  ->  { k }  =  { V } )
54difeq2d 3410 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  ( { V }  \  { V } ) )
6 difid 3641 . . . . . . . . . 10  |-  ( { V }  \  { V } )  =  (/)
75, 6syl6eq 2437 . . . . . . . . 9  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  (/) )
8 preq2 3829 . . . . . . . . . . . 12  |-  ( k  =  V  ->  { x ,  k }  =  { x ,  V } )
98preq1d 3834 . . . . . . . . . . 11  |-  ( k  =  V  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  V } ,  { x ,  l } }
)
109sseq1d 3320 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  V } ,  {
x ,  l } }  C_  ran  E ) )
1110reubidv 2837 . . . . . . . . 9  |-  ( k  =  V  ->  ( E! x  e.  { V }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E  <->  E! x  e.  { V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E ) )
127, 11raleqbidv 2861 . . . . . . . 8  |-  ( k  =  V  ->  ( A. l  e.  ( { V }  \  {
k } ) E! x  e.  { V }  { { x ,  k } ,  {
x ,  l } }  C_  ran  E  <->  A. l  e.  (/)  E! x  e. 
{ V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E ) )
1312ralsng 3791 . . . . . . 7  |-  ( V  e.  X  ->  ( A. k  e.  { V } A. l  e.  ( { V }  \  { k } ) E! x  e.  { V }  { { x ,  k } ,  { x ,  l } }  C_  ran  E  <->  A. l  e.  (/)  E! x  e.  { V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E ) )
1413adantl 453 . . . . . 6  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  ( A. k  e. 
{ V } A. l  e.  ( { V }  \  { k } ) E! x  e.  { V }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  A. l  e.  (/)  E! x  e.  { V }  { { x ,  V } ,  {
x ,  l } }  C_  ran  E ) )
153, 14mpbiri 225 . . . . 5  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  A. k  e.  { V } A. l  e.  ( { V }  \  { k } ) E! x  e.  { V }  { { x ,  k } ,  { x ,  l } }  C_  ran  E )
16 isfrgra 27745 . . . . . 6  |-  ( ( { V }  e.  _V  /\  E  e.  _V )  ->  ( { V } FriendGrph  E  <->  ( { V } USGrph  E  /\  A. k  e.  { V } A. l  e.  ( { V }  \  { k } ) E! x  e.  { V }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E ) ) )
1716ad2antrr 707 . . . . 5  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  ( { V } FriendGrph  E  <-> 
( { V } USGrph  E  /\  A. k  e. 
{ V } A. l  e.  ( { V }  \  { k } ) E! x  e.  { V }  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E ) ) )
182, 15, 17mpbir2and 889 . . . 4  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } FriendGrph  E )
1918ex 424 . . 3  |-  ( ( ( { V }  e.  _V  /\  E  e. 
_V )  /\  { V } USGrph  E )  -> 
( V  e.  X  ->  { V } FriendGrph  E ) )
201, 19mpancom 651 . 2  |-  ( { V } USGrph  E  ->  ( V  e.  X  ->  { V } FriendGrph  E ) )
2120impcom 420 1  |-  ( ( V  e.  X  /\  { V } USGrph  E )  ->  { V } FriendGrph  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2651   E!wreu 2653   _Vcvv 2901    \ cdif 3262    C_ wss 3265   (/)c0 3573   {csn 3759   {cpr 3760   class class class wbr 4155   ran crn 4821   USGrph cusg 21234   FriendGrph cfrgra 27743
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-dm 4830  df-rn 4831  df-usgra 21236  df-frgra 27744
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