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Theorem frsn 5059
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 4827 . . . 4  |-  ( R  Fr  { A }  <->  A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2 df-ne 2651 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
3 simpr 459 . . . . . . . . . . . 12  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  x  C_ 
{ A } )
4 sssn 4174 . . . . . . . . . . . 12  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
53, 4sylib 196 . . . . . . . . . . 11  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =  (/)  \/  x  =  { A } ) )
65ord 375 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  ( -.  x  =  (/)  ->  x  =  { A } ) )
72, 6syl5bi 217 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =/=  (/)  ->  x  =  { A } ) )
87impr 617 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  ( x  C_  { A }  /\  x  =/=  (/) ) )  ->  x  =  { A } )
9 eqimss 3541 . . . . . . . . . 10  |-  ( x  =  { A }  ->  x  C_  { A } )
109adantl 464 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  C_ 
{ A } )
11 simpr 459 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =  { A } )
12 snnzg 4133 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
1312ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  { A }  =/=  (/) )
1411, 13eqnetrd 2747 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =/=  (/) )
1510, 14jca 530 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  (
x  C_  { A }  /\  x  =/=  (/) ) )
168, 15impbida 830 . . . . . . 7  |-  ( ( Rel  R  /\  A  e.  _V )  ->  (
( x  C_  { A }  /\  x  =/=  (/) )  <->  x  =  { A } ) )
1716imbi1d 315 . . . . . 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 1718 . . . . 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 4678 . . . . . 6  |-  { A }  e.  _V
20 raleq 3051 . . . . . . 7  |-  ( x  =  { A }  ->  ( A. z  e.  x  -.  z R y  <->  A. z  e.  { A }  -.  z R y ) )
2120rexeqbi1dv 3060 . . . . . 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 3134 . . . . 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 261 . . . 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 257 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  E. y  e.  { A } A. z  e. 
{ A }  -.  z R y ) )
25 breq2 4443 . . . . . . . 8  |-  ( y  =  A  ->  (
z R y  <->  z R A ) )
2625notbid 292 . . . . . . 7  |-  ( y  =  A  ->  ( -.  z R y  <->  -.  z R A ) )
2726ralbidv 2893 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
2827rexsng 4052 . . . . 5  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
29 breq1 4442 . . . . . . 7  |-  ( z  =  A  ->  (
z R A  <->  A R A ) )
3029notbid 292 . . . . . 6  |-  ( z  =  A  ->  ( -.  z R A  <->  -.  A R A ) )
3130ralsng 4051 . . . . 5  |-  ( A  e.  _V  ->  ( A. z  e.  { A }  -.  z R A  <->  -.  A R A ) )
3228, 31bitrd 253 . . . 4  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3332adantl 464 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3424, 33bitrd 253 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  -.  A R A ) )
35 snprc 4079 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
36 fr0 4847 . . . . . 6  |-  R  Fr  (/)
37 freq2 4839 . . . . . 6  |-  ( { A }  =  (/)  ->  ( R  Fr  { A }  <->  R  Fr  (/) ) )
3836, 37mpbiri 233 . . . . 5  |-  ( { A }  =  (/)  ->  R  Fr  { A } )
3935, 38sylbi 195 . . . 4  |-  ( -.  A  e.  _V  ->  R  Fr  { A }
)
4039adantl 464 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  R  Fr  { A } )
41 brrelex 5027 . . . 4  |-  ( ( Rel  R  /\  A R A )  ->  A  e.  _V )
4241stoic1a 1609 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R A )
4340, 422thd 240 . 2  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R  Fr  { A }  <->  -.  A R A ) )
4434, 43pm2.61dan 789 1  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367   A.wal 1396    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   _Vcvv 3106    C_ wss 3461   (/)c0 3783   {csn 4016   class class class wbr 4439    Fr wfr 4824   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-fr 4827  df-xp 4994  df-rel 4995
This theorem is referenced by:  wesn  5060
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