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Theorem brsset 28056
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 28055 . . 3  |-  Rel  SSet
21brrelexi 4979 . 2  |-  ( A
SSet B  ->  A  e. 
_V )
3 brsset.1 . . 3  |-  B  e. 
_V
43ssex 4536 . 2  |-  ( A 
C_  B  ->  A  e.  _V )
5 breq1 4395 . . 3  |-  ( x  =  A  ->  (
x SSet B  <->  A SSet B ) )
6 sseq1 3477 . . 3  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
7 opex 4656 . . . . . . 7  |-  <. x ,  B >.  e.  _V
87elrn 5180 . . . . . 6  |-  ( <.
x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  E. y  y (  _E  (x)  ( _V  \  _E  ) ) <.
x ,  B >. )
9 vex 3073 . . . . . . . . 9  |-  y  e. 
_V
10 vex 3073 . . . . . . . . 9  |-  x  e. 
_V
119, 10, 3brtxp 28047 . . . . . . . 8  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  _E  x  /\  y ( _V  \  _E  ) B ) )
12 epel 4735 . . . . . . . . 9  |-  ( y  _E  x  <->  y  e.  x )
13 brv 28044 . . . . . . . . . . 11  |-  y _V B
14 brdif 4442 . . . . . . . . . . 11  |-  ( y ( _V  \  _E  ) B  <->  ( y _V B  /\  -.  y  _E  B ) )
1513, 14mpbiran 909 . . . . . . . . . 10  |-  ( y ( _V  \  _E  ) B  <->  -.  y  _E  B )
163epelc 4734 . . . . . . . . . 10  |-  ( y  _E  B  <->  y  e.  B )
1715, 16xchbinx 310 . . . . . . . . 9  |-  ( y ( _V  \  _E  ) B  <->  -.  y  e.  B )
1812, 17anbi12i 697 . . . . . . . 8  |-  ( ( y  _E  x  /\  y ( _V  \  _E  ) B )  <->  ( y  e.  x  /\  -.  y  e.  B ) )
1911, 18bitri 249 . . . . . . 7  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  e.  x  /\  -.  y  e.  B
) )
2019exbii 1635 . . . . . 6  |-  ( E. y  y (  _E 
(x)  ( _V  \  _E  ) ) <. x ,  B >.  <->  E. y ( y  e.  x  /\  -.  y  e.  B )
)
21 exanali 1638 . . . . . 6  |-  ( E. y ( y  e.  x  /\  -.  y  e.  B )  <->  -.  A. y
( y  e.  x  ->  y  e.  B ) )
228, 20, 213bitrri 272 . . . . 5  |-  ( -. 
A. y ( y  e.  x  ->  y  e.  B )  <->  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
2322con1bii 331 . . . 4  |-  ( -. 
<. x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  A. y ( y  e.  x  ->  y  e.  B ) )
24 df-br 4393 . . . . 5  |-  ( x
SSet B  <->  <. x ,  B >.  e.  SSet )
25 df-sset 28022 . . . . . . 7  |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
2625eleq2i 2529 . . . . . 6  |-  ( <.
x ,  B >.  e. 
SSet 
<-> 
<. x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) ) )
2710, 3opelvv 4985 . . . . . . 7  |-  <. x ,  B >.  e.  ( _V  X.  _V )
28 eldif 3438 . . . . . . 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 909 . . . . . 6  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
3026, 29bitri 249 . . . . 5  |-  ( <.
x ,  B >.  e. 
SSet 
<->  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) )
3124, 30bitri 249 . . . 4  |-  ( x
SSet B  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
32 dfss2 3445 . . . 4  |-  ( x 
C_  B  <->  A. y
( y  e.  x  ->  y  e.  B ) )
3323, 31, 323bitr4i 277 . . 3  |-  ( x
SSet B  <->  x  C_  B )
345, 6, 33vtoclbg 3129 . 2  |-  ( A  e.  _V  ->  ( A SSet B  <->  A  C_  B
) )
352, 4, 34pm5.21nii 353 1  |-  ( A
SSet B  <->  A  C_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368   E.wex 1587    e. wcel 1758   _Vcvv 3070    \ cdif 3425    C_ wss 3428   <.cop 3983   class class class wbr 4392    _E cep 4730    X. cxp 4938   ran crn 4941    (x) ctxp 27996   SSetcsset 27998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-eprel 4732  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-1st 6679  df-2nd 6680  df-txp 28020  df-sset 28022
This theorem is referenced by:  idsset  28057  dfon3  28059  imagesset  28120
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