MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfrgra Structured version   Unicode version

Theorem isfrgra 25117
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 4459 . . 3  |-  ( v  =  V  ->  (
v USGrph  e  <->  V USGrph  e )
)
2 difeq1 3611 . . . . 5  |-  ( v  =  V  ->  (
v  \  { k } )  =  ( V  \  { k } ) )
3 reueq1 3056 . . . . 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 3068 . . . 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 3062 . . 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 4460 . . 3  |-  ( e  =  E  ->  ( V USGrph  e  <->  V USGrph  E ) )
8 rneq 5238 . . . . . 6  |-  ( e  =  E  ->  ran  e  =  ran  E )
98sseq2d 3527 . . . . 5  |-  ( e  =  E  ->  ( { { x ,  k } ,  { x ,  l } }  C_ 
ran  e  <->  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) )
109reubidv 3042 . . . 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 2901 . . 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 25116 . 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 4775 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 1395    e. wcel 1819   A.wral 2807   E!wreu 2809    \ cdif 3468    C_ wss 3471   {csn 4032   {cpr 4034   class class class wbr 4456   ran crn 5009   USGrph cusg 24457   FriendGrph cfrgra 25115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-reu 2814  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-cnv 5016  df-dm 5018  df-rn 5019  df-frgra 25116
This theorem is referenced by:  frisusgrapr  25118  frgra0v  25120  frgra1v  25125  frgra2v  25126  frgra3v  25129
  Copyright terms: Public domain W3C validator