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Theorem frgra1v 28315
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 21363 . . 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 3724 . . . . . 6  |-  A. l  e.  (/)  E! x  e. 
{ V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E
4 sneq 3817 . . . . . . . . . . 11  |-  ( k  =  V  ->  { k }  =  { V } )
54difeq2d 3457 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  ( { V }  \  { V } ) )
6 difid 3688 . . . . . . . . . 10  |-  ( { V }  \  { V } )  =  (/)
75, 6syl6eq 2483 . . . . . . . . 9  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  (/) )
8 preq2 3876 . . . . . . . . . . . 12  |-  ( k  =  V  ->  { x ,  k }  =  { x ,  V } )
98preq1d 3881 . . . . . . . . . . 11  |-  ( k  =  V  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  V } ,  { x ,  l } }
)
109sseq1d 3367 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  V } ,  {
x ,  l } }  C_  ran  E ) )
1110reubidv 2884 . . . . . . . . 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 2908 . . . . . . . 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 3838 . . . . . . 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 28307 . . . . . 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 1652    e. wcel 1725   A.wral 2697   E!wreu 2699   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   {csn 3806   {cpr 3807   class class class wbr 4204   ran crn 4871   USGrph cusg 21357   FriendGrph cfrgra 28305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-usgra 21359  df-frgra 28306
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