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Theorem frgra1v 25296
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 24636 . . 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 3877 . . . . . 6  |-  A. l  e.  (/)  E! x  e. 
{ V }  { { x ,  V } ,  { x ,  l } }  C_ 
ran  E
4 sneq 3981 . . . . . . . . . . 11  |-  ( k  =  V  ->  { k }  =  { V } )
54difeq2d 3560 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  ( { V }  \  { V } ) )
6 difid 3839 . . . . . . . . . 10  |-  ( { V }  \  { V } )  =  (/)
75, 6syl6eq 2459 . . . . . . . . 9  |-  ( k  =  V  ->  ( { V }  \  {
k } )  =  (/) )
8 preq2 4051 . . . . . . . . . . . 12  |-  ( k  =  V  ->  { x ,  k }  =  { x ,  V } )
98preq1d 4056 . . . . . . . . . . 11  |-  ( k  =  V  ->  { {
x ,  k } ,  { x ,  l } }  =  { { x ,  V } ,  { x ,  l } }
)
109sseq1d 3468 . . . . . . . . . 10  |-  ( k  =  V  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  E  <->  { { x ,  V } ,  {
x ,  l } }  C_  ran  E ) )
1110reubidv 2991 . . . . . . . . 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 3017 . . . . . . . 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 4006 . . . . . . 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 464 . . . . . 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 25288 . . . . . 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 724 . . . . 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 923 . . . 4  |-  ( ( ( ( { V }  e.  _V  /\  E  e.  _V )  /\  { V } USGrph  E )  /\  V  e.  X )  ->  { V } FriendGrph  E )
1918ex 432 . . 3  |-  ( ( ( { V }  e.  _V  /\  E  e. 
_V )  /\  { V } USGrph  E )  -> 
( V  e.  X  ->  { V } FriendGrph  E ) )
201, 19mpancom 667 . 2  |-  ( { V } USGrph  E  ->  ( V  e.  X  ->  { V } FriendGrph  E ) )
2120impcom 428 1  |-  ( ( V  e.  X  /\  { V } USGrph  E )  ->  { V } FriendGrph  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E!wreu 2755   _Vcvv 3058    \ cdif 3410    C_ wss 3413   (/)c0 3737   {csn 3971   {cpr 3973   class class class wbr 4394   ran crn 4943   USGrph cusg 24628   FriendGrph cfrgra 25286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4948  df-rel 4949  df-cnv 4950  df-dm 4952  df-rn 4953  df-usgra 24631  df-frgra 25287
This theorem is referenced by: (None)
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