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Theorem frsn 4760
Description: Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
frsn  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )

Proof of Theorem frsn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 4352 . . . 4  |-  ( R  Fr  { A }  <->  A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2 df-ne 2448 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
3 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  x  C_ 
{ A } )
4 sssn 3772 . . . . . . . . . . . 12  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
53, 4sylib 188 . . . . . . . . . . 11  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =  (/)  \/  x  =  { A } ) )
65ord 366 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  ( -.  x  =  (/)  ->  x  =  { A } ) )
72, 6syl5bi 208 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =/=  (/)  ->  x  =  { A } ) )
87impr 602 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  ( x  C_  { A }  /\  x  =/=  (/) ) )  ->  x  =  { A } )
9 eqimss 3230 . . . . . . . . . 10  |-  ( x  =  { A }  ->  x  C_  { A } )
109adantl 452 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  C_ 
{ A } )
11 simpr 447 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =  { A } )
12 snnzg 3743 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
1312ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  { A }  =/=  (/) )
1411, 13eqnetrd 2464 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =/=  (/) )
1510, 14jca 518 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  (
x  C_  { A }  /\  x  =/=  (/) ) )
168, 15impbida 805 . . . . . . 7  |-  ( ( Rel  R  /\  A  e.  _V )  ->  (
( x  C_  { A }  /\  x  =/=  (/) )  <->  x  =  { A } ) )
1716imbi1d 308 . . . . . 6  |-  ( ( Rel  R  /\  A  e.  _V )  ->  (
( ( x  C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  ( x  =  { A }  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) ) )
1817albidv 1611 . . . . 5  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  A. x ( x  =  { A }  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) ) )
19 snex 4216 . . . . . 6  |-  { A }  e.  _V
20 raleq 2736 . . . . . . 7  |-  ( x  =  { A }  ->  ( A. z  e.  x  -.  z R y  <->  A. z  e.  { A }  -.  z R y ) )
2120rexeqbi1dv 2745 . . . . . 6  |-  ( x  =  { A }  ->  ( E. y  e.  x  A. z  e.  x  -.  z R y  <->  E. y  e.  { A } A. z  e. 
{ A }  -.  z R y ) )
2219, 21ceqsalv 2814 . . . . 5  |-  ( A. x ( x  =  { A }  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  E. y  e.  { A } A. z  e.  { A }  -.  z R y )
2318, 22syl6bb 252 . . . 4  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y )  <->  E. y  e.  { A } A. z  e.  { A }  -.  z R y ) )
241, 23syl5bb 248 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  E. y  e.  { A } A. z  e. 
{ A }  -.  z R y ) )
25 breq2 4027 . . . . . . . 8  |-  ( y  =  A  ->  (
z R y  <->  z R A ) )
2625notbid 285 . . . . . . 7  |-  ( y  =  A  ->  ( -.  z R y  <->  -.  z R A ) )
2726ralbidv 2563 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
2827rexsng 3673 . . . . 5  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
29 breq1 4026 . . . . . . 7  |-  ( z  =  A  ->  (
z R A  <->  A R A ) )
3029notbid 285 . . . . . 6  |-  ( z  =  A  ->  ( -.  z R A  <->  -.  A R A ) )
3130ralsng 3672 . . . . 5  |-  ( A  e.  _V  ->  ( A. z  e.  { A }  -.  z R A  <->  -.  A R A ) )
3228, 31bitrd 244 . . . 4  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3332adantl 452 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3424, 33bitrd 244 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  -.  A R A ) )
35 snprc 3695 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
36 fr0 4372 . . . . . 6  |-  R  Fr  (/)
37 freq2 4364 . . . . . 6  |-  ( { A }  =  (/)  ->  ( R  Fr  { A }  <->  R  Fr  (/) ) )
3836, 37mpbiri 224 . . . . 5  |-  ( { A }  =  (/)  ->  R  Fr  { A } )
3935, 38sylbi 187 . . . 4  |-  ( -.  A  e.  _V  ->  R  Fr  { A }
)
4039adantl 452 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  R  Fr  { A } )
41 brrelex 4727 . . . . 5  |-  ( ( Rel  R  /\  A R A )  ->  A  e.  _V )
4241ex 423 . . . 4  |-  ( Rel 
R  ->  ( A R A  ->  A  e. 
_V ) )
4342con3and 428 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R A )
4440, 432thd 231 . 2  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R  Fr  { A }  <->  -.  A R A ) )
4534, 44pm2.61dan 766 1  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    Fr wfr 4349   Rel wrel 4694
This theorem is referenced by:  wesn  4761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-fr 4352  df-xp 4695  df-rel 4696
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