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Theorem brcap 25503
Description: Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcap.1  |-  A  e. 
_V
brcap.2  |-  B  e. 
_V
brcap.3  |-  C  e. 
_V
Assertion
Ref Expression
brcap  |-  ( <. A ,  B >.Cap C  <-> 
C  =  ( A  i^i  B ) )

Proof of Theorem brcap
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4368 . 2  |-  <. A ,  B >.  e.  _V
2 brcap.3 . 2  |-  C  e. 
_V
3 df-cap 25435 . 2  |- Cap  =  ( ( ( _V  X.  _V )  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (
( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  )
)  (x)  _V )
) )
4 brcap.1 . . . 4  |-  A  e. 
_V
5 brcap.2 . . . 4  |-  B  e. 
_V
64, 5opelvv 4864 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4849 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 887 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 vex 2902 . . . . . . 7  |-  x  e. 
_V
109, 1brco 4983 . . . . . 6  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. ) )
11 ancom 438 . . . . . . . 8  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y `' 1st <. A ,  B >.  /\  x  _E  y
) )
12 vex 2902 . . . . . . . . . . 11  |-  y  e. 
_V
1312, 1brcnv 4995 . . . . . . . . . 10  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
144, 5, 12br1steq 25154 . . . . . . . . . 10  |-  ( <. A ,  B >. 1st y  <->  y  =  A )
1513, 14bitri 241 . . . . . . . . 9  |-  ( y `' 1st <. A ,  B >.  <-> 
y  =  A )
1615anbi1i 677 . . . . . . . 8  |-  ( ( y `' 1st <. A ,  B >.  /\  x  _E  y )  <->  ( y  =  A  /\  x  _E  y ) )
1711, 16bitri 241 . . . . . . 7  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y  =  A  /\  x  _E  y ) )
1817exbii 1589 . . . . . 6  |-  ( E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  E. y ( y  =  A  /\  x  _E  y ) )
1910, 18bitri 241 . . . . 5  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( y  =  A  /\  x  _E  y ) )
20 breq2 4157 . . . . . 6  |-  ( y  =  A  ->  (
x  _E  y  <->  x  _E  A ) )
214, 20ceqsexv 2934 . . . . 5  |-  ( E. y ( y  =  A  /\  x  _E  y )  <->  x  _E  A )
224epelc 4437 . . . . 5  |-  ( x  _E  A  <->  x  e.  A )
2319, 21, 223bitri 263 . . . 4  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <-> 
x  e.  A )
249, 1brco 4983 . . . . . 6  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. ) )
25 ancom 438 . . . . . . . 8  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y `' 2nd <. A ,  B >.  /\  x  _E  y
) )
2612, 1brcnv 4995 . . . . . . . . . 10  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
274, 5, 12br2ndeq 25155 . . . . . . . . . 10  |-  ( <. A ,  B >. 2nd y  <->  y  =  B )
2826, 27bitri 241 . . . . . . . . 9  |-  ( y `' 2nd <. A ,  B >.  <-> 
y  =  B )
2928anbi1i 677 . . . . . . . 8  |-  ( ( y `' 2nd <. A ,  B >.  /\  x  _E  y )  <->  ( y  =  B  /\  x  _E  y ) )
3025, 29bitri 241 . . . . . . 7  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y  =  B  /\  x  _E  y ) )
3130exbii 1589 . . . . . 6  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  _E  y ) )
3224, 31bitri 241 . . . . 5  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( y  =  B  /\  x  _E  y ) )
33 breq2 4157 . . . . . 6  |-  ( y  =  B  ->  (
x  _E  y  <->  x  _E  B ) )
345, 33ceqsexv 2934 . . . . 5  |-  ( E. y ( y  =  B  /\  x  _E  y )  <->  x  _E  B )
355epelc 4437 . . . . 5  |-  ( x  _E  B  <->  x  e.  B )
3632, 34, 353bitri 263 . . . 4  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <-> 
x  e.  B )
3723, 36anbi12i 679 . . 3  |-  ( ( x ( `' 1st  o.  _E  ) <. A ,  B >.  /\  x ( `' 2nd  o.  _E  ) <. A ,  B >. )  <-> 
( x  e.  A  /\  x  e.  B
) )
38 brin 4200 . . 3  |-  ( x ( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  )
) <. A ,  B >.  <-> 
( x ( `' 1st  o.  _E  ) <. A ,  B >.  /\  x ( `' 2nd  o.  _E  ) <. A ,  B >. ) )
39 elin 3473 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
4037, 38, 393bitr4ri 270 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x ( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  ) ) <. A ,  B >. )
411, 2, 3, 8, 40brtxpsd3 25460 1  |-  ( <. A ,  B >.Cap C  <-> 
C  =  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899    i^i cin 3262   <.cop 3760   class class class wbr 4153    _E cep 4433    X. cxp 4816   `'ccnv 4817    o. ccom 4822   1stc1st 6286   2ndc2nd 6287  Capccap 25414
This theorem is referenced by:  brrestrict  25512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-eprel 4435  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-1st 6288  df-2nd 6289  df-symdif 25386  df-txp 25419  df-cap 25435
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