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Theorem brsset 30704
Description: For sets, the  SSet binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypothesis
Ref Expression
brsset.1  |-  B  e. 
_V
Assertion
Ref Expression
brsset  |-  ( A
SSet B  <->  A  C_  B )

Proof of Theorem brsset
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsset 30703 . . 3  |-  Rel  SSet
21brrelexi 4893 . 2  |-  ( A
SSet B  ->  A  e. 
_V )
3 brsset.1 . . 3  |-  B  e. 
_V
43ssex 4560 . 2  |-  ( A 
C_  B  ->  A  e.  _V )
5 breq1 4418 . . 3  |-  ( x  =  A  ->  (
x SSet B  <->  A SSet B ) )
6 sseq1 3464 . . 3  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
7 opex 4677 . . . . . . 7  |-  <. x ,  B >.  e.  _V
87elrn 5093 . . . . . 6  |-  ( <.
x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  E. y  y (  _E  (x)  ( _V  \  _E  ) ) <.
x ,  B >. )
9 vex 3059 . . . . . . . . 9  |-  y  e. 
_V
10 vex 3059 . . . . . . . . 9  |-  x  e. 
_V
119, 10, 3brtxp 30695 . . . . . . . 8  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  _E  x  /\  y ( _V  \  _E  ) B ) )
12 epel 4766 . . . . . . . . 9  |-  ( y  _E  x  <->  y  e.  x )
13 brv 30692 . . . . . . . . . . 11  |-  y _V B
14 brdif 4466 . . . . . . . . . . 11  |-  ( y ( _V  \  _E  ) B  <->  ( y _V B  /\  -.  y  _E  B ) )
1513, 14mpbiran 934 . . . . . . . . . 10  |-  ( y ( _V  \  _E  ) B  <->  -.  y  _E  B )
163epelc 4765 . . . . . . . . . 10  |-  ( y  _E  B  <->  y  e.  B )
1715, 16xchbinx 316 . . . . . . . . 9  |-  ( y ( _V  \  _E  ) B  <->  -.  y  e.  B )
1812, 17anbi12i 708 . . . . . . . 8  |-  ( ( y  _E  x  /\  y ( _V  \  _E  ) B )  <->  ( y  e.  x  /\  -.  y  e.  B ) )
1911, 18bitri 257 . . . . . . 7  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  e.  x  /\  -.  y  e.  B
) )
2019exbii 1728 . . . . . 6  |-  ( E. y  y (  _E 
(x)  ( _V  \  _E  ) ) <. x ,  B >.  <->  E. y ( y  e.  x  /\  -.  y  e.  B )
)
21 exanali 1731 . . . . . 6  |-  ( E. y ( y  e.  x  /\  -.  y  e.  B )  <->  -.  A. y
( y  e.  x  ->  y  e.  B ) )
228, 20, 213bitrri 280 . . . . 5  |-  ( -. 
A. y ( y  e.  x  ->  y  e.  B )  <->  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
2322con1bii 337 . . . 4  |-  ( -. 
<. x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  A. y ( y  e.  x  ->  y  e.  B ) )
24 df-br 4416 . . . . 5  |-  ( x
SSet B  <->  <. x ,  B >.  e.  SSet )
25 df-sset 30670 . . . . . . 7  |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
2625eleq2i 2531 . . . . . 6  |-  ( <.
x ,  B >.  e. 
SSet 
<-> 
<. x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) ) )
2710, 3opelvv 4899 . . . . . . 7  |-  <. x ,  B >.  e.  ( _V  X.  _V )
28 eldif 3425 . . . . . . 7  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  ( <. x ,  B >.  e.  ( _V  X.  _V )  /\  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) ) )
2927, 28mpbiran 934 . . . . . 6  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
3026, 29bitri 257 . . . . 5  |-  ( <.
x ,  B >.  e. 
SSet 
<->  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) )
3124, 30bitri 257 . . . 4  |-  ( x
SSet B  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
32 dfss2 3432 . . . 4  |-  ( x 
C_  B  <->  A. y
( y  e.  x  ->  y  e.  B ) )
3323, 31, 323bitr4i 285 . . 3  |-  ( x
SSet B  <->  x  C_  B )
345, 6, 33vtoclbg 3119 . 2  |-  ( A  e.  _V  ->  ( A SSet B  <->  A  C_  B
) )
352, 4, 34pm5.21nii 359 1  |-  ( A
SSet B  <->  A  C_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673    e. wcel 1897   _Vcvv 3056    \ cdif 3412    C_ wss 3415   <.cop 3985   class class class wbr 4415    _E cep 4761    X. cxp 4850   ran crn 4853    (x) ctxp 30644   SSetcsset 30646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-eprel 4763  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-1st 6819  df-2nd 6820  df-txp 30668  df-sset 30670
This theorem is referenced by:  idsset  30705  dfon3  30707  imagesset  30768
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