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Theorem frgra1v 30725
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 23402 . . 3  |-  ( { V } USGrph  E  ->  ( { V }  e.  _V  /\  E  e.  _V ) )
2 simplr 754 . . . . 5  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } USGrph  E )
3 ral0 3879 . . . . . 6  |-  A. l  e.  (/)  E! x  e. 
{ V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E
4 sneq 3982 . . . . . . . . . . 11  |-  ( k  =  V  ->  { k }  =  { V } )
54difeq2d 3569 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  ( { V }  \  { V } ) )
6 difid 3842 . . . . . . . . . 10  |-  ( { V }  \  { V } )  =  (/)
75, 6syl6eq 2507 . . . . . . . . 9  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  (/) )
8 preq2 4050 . . . . . . . . . . . 12  |-  ( k  =  V  ->  { x ,  k }  =  { x ,  V } )
98preq1d 4055 . . . . . . . . . . 11  |-  ( k  =  V  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  V } ,  { x ,  l } }
)
109sseq1d 3478 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  V } ,  {
x ,  l } }  C_  ran  E ) )
1110reubidv 2998 . . . . . . . . 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 3024 . . . . . . . 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 4007 . . . . . . 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 466 . . . . . 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 233 . . . . 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 30717 . . . . . 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 725 . . . . 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 913 . . . 4  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } FriendGrph  E )
1918ex 434 . . 3  |-  ( ( ( { V }  e.  _V  /\  E  e. 
_V )  /\  { V } USGrph  E )  -> 
( V  e.  X  ->  { V } FriendGrph  E ) )
201, 19mpancom 669 . 2  |-  ( { V } USGrph  E  ->  ( V  e.  X  ->  { V } FriendGrph  E ) )
2120impcom 430 1  |-  ( ( V  e.  X  /\  { V } USGrph  E )  ->  { V } FriendGrph  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793   E!wreu 2795   _Vcvv 3065    \ cdif 3420    C_ wss 3423   (/)c0 3732   {csn 3972   {cpr 3974   class class class wbr 4387   ran crn 4936   USGrph cusg 23396   FriendGrph cfrgra 30715
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-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-br 4388  df-opab 4446  df-xp 4941  df-rel 4942  df-cnv 4943  df-dm 4945  df-rn 4946  df-usgra 23398  df-frgra 30716
This theorem is referenced by: (None)
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