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Theorem frgra1v 28422
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 28233 . . 3  |-  ( { V } USGrph  E  ->  ( { V }  e.  _V  /\  E  e.  _V ) )
2 simplr 731 . . . . 5  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } USGrph  E )
3 ral0 3571 . . . . . 6  |-  A. l  e.  (/)  E! x  e. 
{ V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E
4 sneq 3664 . . . . . . . . . . 11  |-  ( k  =  V  ->  { k }  =  { V } )
54difeq2d 3307 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  ( { V }  \  { V } ) )
6 difid 3535 . . . . . . . . . 10  |-  ( { V }  \  { V } )  =  (/)
75, 6syl6eq 2344 . . . . . . . . 9  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  (/) )
8 preq2 3720 . . . . . . . . . . . 12  |-  ( k  =  V  ->  { x ,  k }  =  { x ,  V } )
98preq1d 3725 . . . . . . . . . . 11  |-  ( k  =  V  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  V } ,  { x ,  l } }
)
109sseq1d 3218 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  V } ,  {
x ,  l } }  C_  ran  E ) )
1110reubidv 2737 . . . . . . . . 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 2761 . . . . . . . 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 3685 . . . . . . 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 452 . . . . . 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 224 . . . . 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 28417 . . . . . 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 706 . . . . 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 888 . . . 4  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } FriendGrph  E )
1918ex 423 . . 3  |-  ( ( ( { V }  e.  _V  /\  E  e. 
_V )  /\  { V } USGrph  E )  -> 
( V  e.  X  ->  { V } FriendGrph  E ) )
201, 19mpancom 650 . 2  |-  ( { V } USGrph  E  ->  ( V  e.  X  ->  { V } FriendGrph  E ) )
2120impcom 419 1  |-  ( ( V  e.  X  /\  { V } USGrph  E )  ->  { V } FriendGrph  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E!wreu 2558   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   class class class wbr 4039   ran crn 4706   USGrph cusg 28227   FriendGrph cfrgra 28415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-usgra 28229  df-frgra 28416
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