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Theorem isfrgra 30720
Description: The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
Assertion
Ref Expression
isfrgra  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) ) )
Distinct variable groups:    k, V, l, x    k, E, l, x
Allowed substitution hints:    X( x, k, l)    Y( x, k, l)

Proof of Theorem isfrgra
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4393 . . 3  |-  ( v  =  V  ->  (
v USGrph  e  <->  V USGrph  e )
)
2 difeq1 3565 . . . . 5  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
3 reueq1 3015 . . . . 5  |-  ( v  =  V  ->  ( E! x  e.  v  { { x ,  k } ,  { x ,  l } }  C_ 
ran  e  <->  E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  e ) )
42, 3raleqbidv 3027 . . . 4  |-  ( v  =  V  ->  ( A. l  e.  (
v  \  { k } ) E! x  e.  v  { { x ,  k } ,  { x ,  l } }  C_  ran  e 
<-> 
A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  e ) )
54raleqbi1dv 3021 . . 3  |-  ( v  =  V  ->  ( A. k  e.  v  A. l  e.  (
v  \  { k } ) E! x  e.  v  { { x ,  k } ,  { x ,  l } }  C_  ran  e 
<-> 
A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  e ) )
61, 5anbi12d 710 . 2  |-  ( v  =  V  ->  (
( v USGrph  e  /\  A. k  e.  v  A. l  e.  ( v  \  { k } ) E! x  e.  v  { { x ,  k } ,  {
x ,  l } }  C_  ran  e )  <-> 
( V USGrph  e  /\  A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  e ) ) )
7 breq2 4394 . . 3  |-  ( e  =  E  ->  ( V USGrph  e  <->  V USGrph  E ) )
8 rneq 5163 . . . . . 6  |-  ( e  =  E  ->  ran  e  =  ran  E )
98sseq2d 3482 . . . . 5  |-  ( e  =  E  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  e  <->  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) )
109reubidv 3001 . . . 4  |-  ( e  =  E  ->  ( E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  e  <->  E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) )
11102ralbidv 2861 . . 3  |-  ( e  =  E  ->  ( A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  e 
<-> 
A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) )
127, 11anbi12d 710 . 2  |-  ( e  =  E  ->  (
( V USGrph  e  /\  A. k  e.  V  A. l  e.  ( V  \  { k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  e )  <->  ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V  \  {
k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_ 
ran  E ) ) )
13 df-frgra 30719 . 2  |- FriendGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v 
A. l  e.  ( v  \  { k } ) E! x  e.  v  { { x ,  k } ,  { x ,  l } }  C_  ran  e ) }
146, 12, 13brabg 4706 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E!wreu 2797    \ cdif 3423    C_ wss 3426   {csn 3975   {cpr 3977   class class class wbr 4390   ran crn 4939   USGrph cusg 23399   FriendGrph cfrgra 30718
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 4511  ax-nul 4519  ax-pr 4629
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 2601  df-ne 2646  df-ral 2800  df-reu 2802  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-cnv 4946  df-dm 4948  df-rn 4949  df-frgra 30719
This theorem is referenced by:  frisusgrapr  30721  frgra0v  30723  frgra1v  30728  frgra2v  30729  frgra3v  30732
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