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Theorem frsn 5056
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 4824 . . . 4  |-  ( R  Fr  { A }  <->  A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2 df-ne 2638 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
3 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  x  C_ 
{ A } )
4 sssn 4169 . . . . . . . . . . . 12  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
53, 4sylib 196 . . . . . . . . . . 11  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =  (/)  \/  x  =  { A } ) )
65ord 377 . . . . . . . . . 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 619 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  ( x  C_  { A }  /\  x  =/=  (/) ) )  ->  x  =  { A } )
9 eqimss 3538 . . . . . . . . . 10  |-  ( x  =  { A }  ->  x  C_  { A } )
109adantl 466 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  C_ 
{ A } )
11 simpr 461 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =  { A } )
12 snnzg 4128 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
1312ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  { A }  =/=  (/) )
1411, 13eqnetrd 2734 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =/=  (/) )
1510, 14jca 532 . . . . . . . 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 317 . . . . . 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 1698 . . . . 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 4674 . . . . . 6  |-  { A }  e.  _V
20 raleq 3038 . . . . . . 7  |-  ( x  =  { A }  ->  ( A. z  e.  x  -.  z R y  <->  A. z  e.  { A }  -.  z R y ) )
2120rexeqbi1dv 3047 . . . . . 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 3121 . . . . 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 4437 . . . . . . . 8  |-  ( y  =  A  ->  (
z R y  <->  z R A ) )
2625notbid 294 . . . . . . 7  |-  ( y  =  A  ->  ( -.  z R y  <->  -.  z R A ) )
2726ralbidv 2880 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
2827rexsng 4046 . . . . 5  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
29 breq1 4436 . . . . . . 7  |-  ( z  =  A  ->  (
z R A  <->  A R A ) )
3029notbid 294 . . . . . 6  |-  ( z  =  A  ->  ( -.  z R A  <->  -.  A R A ) )
3130ralsng 4045 . . . . 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 466 . . 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 4074 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
36 fr0 4844 . . . . . 6  |-  R  Fr  (/)
37 freq2 4836 . . . . . 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 466 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  R  Fr  { A } )
41 brrelex 5024 . . . 4  |-  ( ( Rel  R  /\  A R A )  ->  A  e.  _V )
4241stoic1a 1590 . . 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 368    /\ wa 369   A.wal 1379    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   E.wrex 2792   _Vcvv 3093    C_ wss 3458   (/)c0 3767   {csn 4010   class class class wbr 4433    Fr wfr 4821   Rel wrel 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-br 4434  df-opab 4492  df-fr 4824  df-xp 4991  df-rel 4992
This theorem is referenced by:  wesn  5057
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