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Theorem brsset 25739
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 25738 . . 3  |-  Rel  SSet
21brrelexi 4921 . 2  |-  ( A
SSet B  ->  A  e. 
_V )
3 brsset.1 . . 3  |-  B  e. 
_V
43ssex 4350 . 2  |-  ( A 
C_  B  ->  A  e.  _V )
5 breq1 4218 . . 3  |-  ( x  =  A  ->  (
x SSet B  <->  A SSet B ) )
6 sseq1 3371 . . 3  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
7 opex 4430 . . . . . . 7  |-  <. x ,  B >.  e.  _V
87elrn 5113 . . . . . 6  |-  ( <.
x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  E. y  y (  _E  (x)  ( _V  \  _E  ) ) <.
x ,  B >. )
9 vex 2961 . . . . . . . . 9  |-  y  e. 
_V
10 vex 2961 . . . . . . . . 9  |-  x  e. 
_V
119, 10, 3brtxp 25730 . . . . . . . 8  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  _E  x  /\  y ( _V  \  _E  ) B ) )
12 epel 4500 . . . . . . . . 9  |-  ( y  _E  x  <->  y  e.  x )
13 brv 25727 . . . . . . . . . . 11  |-  y _V B
14 brdif 4263 . . . . . . . . . . 11  |-  ( y ( _V  \  _E  ) B  <->  ( y _V B  /\  -.  y  _E  B ) )
1513, 14mpbiran 886 . . . . . . . . . 10  |-  ( y ( _V  \  _E  ) B  <->  -.  y  _E  B )
163epelc 4499 . . . . . . . . . 10  |-  ( y  _E  B  <->  y  e.  B )
1715, 16xchbinx 303 . . . . . . . . 9  |-  ( y ( _V  \  _E  ) B  <->  -.  y  e.  B )
1812, 17anbi12i 680 . . . . . . . 8  |-  ( ( y  _E  x  /\  y ( _V  \  _E  ) B )  <->  ( y  e.  x  /\  -.  y  e.  B ) )
1911, 18bitri 242 . . . . . . 7  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  e.  x  /\  -.  y  e.  B
) )
2019exbii 1593 . . . . . 6  |-  ( E. y  y (  _E 
(x)  ( _V  \  _E  ) ) <. x ,  B >.  <->  E. y ( y  e.  x  /\  -.  y  e.  B )
)
21 exanali 1596 . . . . . 6  |-  ( E. y ( y  e.  x  /\  -.  y  e.  B )  <->  -.  A. y
( y  e.  x  ->  y  e.  B ) )
228, 20, 213bitrri 265 . . . . 5  |-  ( -. 
A. y ( y  e.  x  ->  y  e.  B )  <->  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
2322con1bii 323 . . . 4  |-  ( -. 
<. x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  A. y ( y  e.  x  ->  y  e.  B ) )
24 df-br 4216 . . . . 5  |-  ( x
SSet B  <->  <. x ,  B >.  e.  SSet )
25 df-sset 25705 . . . . . . 7  |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
2625eleq2i 2502 . . . . . 6  |-  ( <.
x ,  B >.  e. 
SSet 
<-> 
<. x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) ) )
2710, 3opelvv 4927 . . . . . . 7  |-  <. x ,  B >.  e.  ( _V  X.  _V )
28 eldif 3332 . . . . . . 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 886 . . . . . 6  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
3026, 29bitri 242 . . . . 5  |-  ( <.
x ,  B >.  e. 
SSet 
<->  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) )
3124, 30bitri 242 . . . 4  |-  ( x
SSet B  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
32 dfss2 3339 . . . 4  |-  ( x 
C_  B  <->  A. y
( y  e.  x  ->  y  e.  B ) )
3323, 31, 323bitr4i 270 . . 3  |-  ( x
SSet B  <->  x  C_  B )
345, 6, 33vtoclbg 3014 . 2  |-  ( A  e.  _V  ->  ( A SSet B  <->  A  C_  B
) )
352, 4, 34pm5.21nii 344 1  |-  ( A
SSet B  <->  A  C_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    e. wcel 1726   _Vcvv 2958    \ cdif 3319    C_ wss 3322   <.cop 3819   class class class wbr 4215    _E cep 4495    X. cxp 4879   ran crn 4882    (x) ctxp 25679   SSetcsset 25681
This theorem is referenced by:  idsset  25740  dfon3  25742  imagesset  25803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-eprel 4497  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-1st 6352  df-2nd 6353  df-txp 25703  df-sset 25705
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