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Theorem frsn 4882
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 4770 . . . 4  |-  ( R  Fr  { A }  <->  A. x ( ( x 
C_  { A }  /\  x  =/=  (/) )  ->  E. y  e.  x  A. z  e.  x  -.  z R y ) )
2 df-ne 2623 . . . . . . . . . 10  |-  ( x  =/=  (/)  <->  -.  x  =  (/) )
3 simpr 467 . . . . . . . . . . . 12  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  x  C_ 
{ A } )
4 sssn 4098 . . . . . . . . . . . 12  |-  ( x 
C_  { A }  <->  ( x  =  (/)  \/  x  =  { A } ) )
53, 4sylib 201 . . . . . . . . . . 11  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =  (/)  \/  x  =  { A } ) )
65ord 383 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  ( -.  x  =  (/)  ->  x  =  { A } ) )
72, 6syl5bi 225 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  C_  { A } )  ->  (
x  =/=  (/)  ->  x  =  { A } ) )
87impr 629 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  ( x  C_  { A }  /\  x  =/=  (/) ) )  ->  x  =  { A } )
9 eqimss 3451 . . . . . . . . . 10  |-  ( x  =  { A }  ->  x  C_  { A } )
109adantl 472 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  C_ 
{ A } )
11 simpr 467 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =  { A } )
12 snnzg 4057 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  { A }  =/=  (/) )
1312ad2antlr 738 . . . . . . . . . 10  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  { A }  =/=  (/) )
1411, 13eqnetrd 2690 . . . . . . . . 9  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  x  =/=  (/) )
1510, 14jca 539 . . . . . . . 8  |-  ( ( ( Rel  R  /\  A  e.  _V )  /\  x  =  { A } )  ->  (
x  C_  { A }  /\  x  =/=  (/) ) )
168, 15impbida 847 . . . . . . 7  |-  ( ( Rel  R  /\  A  e.  _V )  ->  (
( x  C_  { A }  /\  x  =/=  (/) )  <->  x  =  { A } ) )
1716imbi1d 323 . . . . . 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 1770 . . . . 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 4613 . . . . . 6  |-  { A }  e.  _V
20 raleq 2954 . . . . . . 7  |-  ( x  =  { A }  ->  ( A. z  e.  x  -.  z R y  <->  A. z  e.  { A }  -.  z R y ) )
2120rexeqbi1dv 2963 . . . . . 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 3042 . . . . 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 269 . . . 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 265 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  E. y  e.  { A } A. z  e. 
{ A }  -.  z R y ) )
25 breq2 4377 . . . . . . . 8  |-  ( y  =  A  ->  (
z R y  <->  z R A ) )
2625notbid 300 . . . . . . 7  |-  ( y  =  A  ->  ( -.  z R y  <->  -.  z R A ) )
2726ralbidv 2809 . . . . . 6  |-  ( y  =  A  ->  ( A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
2827rexsng 3974 . . . . 5  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  A. z  e.  { A }  -.  z R A ) )
29 breq1 4376 . . . . . . 7  |-  ( z  =  A  ->  (
z R A  <->  A R A ) )
3029notbid 300 . . . . . 6  |-  ( z  =  A  ->  ( -.  z R A  <->  -.  A R A ) )
3130ralsng 3973 . . . . 5  |-  ( A  e.  _V  ->  ( A. z  e.  { A }  -.  z R A  <->  -.  A R A ) )
3228, 31bitrd 261 . . . 4  |-  ( A  e.  _V  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3332adantl 472 . . 3  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( E. y  e.  { A } A. z  e.  { A }  -.  z R y  <->  -.  A R A ) )
3424, 33bitrd 261 . 2  |-  ( ( Rel  R  /\  A  e.  _V )  ->  ( R  Fr  { A } 
<->  -.  A R A ) )
35 snprc 4003 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
36 fr0 4790 . . . . . 6  |-  R  Fr  (/)
37 freq2 4782 . . . . . 6  |-  ( { A }  =  (/)  ->  ( R  Fr  { A }  <->  R  Fr  (/) ) )
3836, 37mpbiri 241 . . . . 5  |-  ( { A }  =  (/)  ->  R  Fr  { A } )
3935, 38sylbi 200 . . . 4  |-  ( -.  A  e.  _V  ->  R  Fr  { A }
)
4039adantl 472 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  R  Fr  { A } )
41 brrelex 4850 . . . 4  |-  ( ( Rel  R  /\  A R A )  ->  A  e.  _V )
4241stoic1a 1658 . . 3  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  -.  A R A )
4340, 422thd 248 . 2  |-  ( ( Rel  R  /\  -.  A  e.  _V )  ->  ( R  Fr  { A }  <->  -.  A R A ) )
4434, 43pm2.61dan 805 1  |-  ( Rel 
R  ->  ( R  Fr  { A }  <->  -.  A R A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375   A.wal 1445    = wceq 1447    e. wcel 1890    =/= wne 2621   A.wral 2736   E.wrex 2737   _Vcvv 3012    C_ wss 3371   (/)c0 3698   {csn 3935   class class class wbr 4373    Fr wfr 4767   Rel wrel 4816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-br 4374  df-opab 4433  df-fr 4770  df-xp 4817  df-rel 4818
This theorem is referenced by:  wesn  4883
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