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Theorem brcap 24479
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 4237 . 2  |-  <. A ,  B >.  e.  _V
2 brcap.3 . 2  |-  C  e. 
_V
3 df-cap 24411 . 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 4735 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4720 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 886 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 vex 2791 . . . . . . 7  |-  x  e. 
_V
109, 1brco 4852 . . . . . 6  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. ) )
11 ancom 437 . . . . . . . 8  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y `' 1st <. A ,  B >.  /\  x  _E  y
) )
12 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
1312, 1brcnv 4864 . . . . . . . . . 10  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
144, 5, 12br1steq 24130 . . . . . . . . . 10  |-  ( <. A ,  B >. 1st y  <->  y  =  A )
1513, 14bitri 240 . . . . . . . . 9  |-  ( y `' 1st <. A ,  B >.  <-> 
y  =  A )
1615anbi1i 676 . . . . . . . 8  |-  ( ( y `' 1st <. A ,  B >.  /\  x  _E  y )  <->  ( y  =  A  /\  x  _E  y ) )
1711, 16bitri 240 . . . . . . 7  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y  =  A  /\  x  _E  y ) )
1817exbii 1569 . . . . . 6  |-  ( E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  E. y ( y  =  A  /\  x  _E  y ) )
1910, 18bitri 240 . . . . 5  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( y  =  A  /\  x  _E  y ) )
20 breq2 4027 . . . . . 6  |-  ( y  =  A  ->  (
x  _E  y  <->  x  _E  A ) )
214, 20ceqsexv 2823 . . . . 5  |-  ( E. y ( y  =  A  /\  x  _E  y )  <->  x  _E  A )
224epelc 4307 . . . . 5  |-  ( x  _E  A  <->  x  e.  A )
2319, 21, 223bitri 262 . . . 4  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <-> 
x  e.  A )
249, 1brco 4852 . . . . . 6  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. ) )
25 ancom 437 . . . . . . . 8  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y `' 2nd <. A ,  B >.  /\  x  _E  y
) )
2612, 1brcnv 4864 . . . . . . . . . 10  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
274, 5, 12br2ndeq 24131 . . . . . . . . . 10  |-  ( <. A ,  B >. 2nd y  <->  y  =  B )
2826, 27bitri 240 . . . . . . . . 9  |-  ( y `' 2nd <. A ,  B >.  <-> 
y  =  B )
2928anbi1i 676 . . . . . . . 8  |-  ( ( y `' 2nd <. A ,  B >.  /\  x  _E  y )  <->  ( y  =  B  /\  x  _E  y ) )
3025, 29bitri 240 . . . . . . 7  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y  =  B  /\  x  _E  y ) )
3130exbii 1569 . . . . . 6  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  _E  y ) )
3224, 31bitri 240 . . . . 5  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( y  =  B  /\  x  _E  y ) )
33 breq2 4027 . . . . . 6  |-  ( y  =  B  ->  (
x  _E  y  <->  x  _E  B ) )
345, 33ceqsexv 2823 . . . . 5  |-  ( E. y ( y  =  B  /\  x  _E  y )  <->  x  _E  B )
355epelc 4307 . . . . 5  |-  ( x  _E  B  <->  x  e.  B )
3632, 34, 353bitri 262 . . . 4  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <-> 
x  e.  B )
3723, 36anbi12i 678 . . 3  |-  ( ( x ( `' 1st  o.  _E  ) <. A ,  B >.  /\  x ( `' 2nd  o.  _E  ) <. A ,  B >. )  <-> 
( x  e.  A  /\  x  e.  B
) )
38 brin 4070 . . 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 3358 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
4037, 38, 393bitr4ri 269 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x ( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  ) ) <. A ,  B >. )
411, 2, 3, 8, 40brtxpsd3 24436 1  |-  ( <. A ,  B >.Cap C  <-> 
C  =  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   <.cop 3643   class class class wbr 4023    _E cep 4303    X. cxp 4687   `'ccnv 4688    o. ccom 4693   1stc1st 6120   2ndc2nd 6121  Capccap 24390
This theorem is referenced by:  brrestrict  24487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 24362  df-txp 24395  df-cap 24411
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