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Theorem brcap 28108
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 4657 . 2  |-  <. A ,  B >.  e.  _V
2 brcap.3 . 2  |-  C  e. 
_V
3 df-cap 28037 . 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 4986 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4971 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 911 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 epel 4736 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
10 vex 3074 . . . . . . . . 9  |-  y  e. 
_V
1110, 1brcnv 5123 . . . . . . . 8  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
124, 5, 10br1steq 27722 . . . . . . . 8  |-  ( <. A ,  B >. 1st y  <->  y  =  A )
1311, 12bitri 249 . . . . . . 7  |-  ( y `' 1st <. A ,  B >.  <-> 
y  =  A )
149, 13anbi12ci 698 . . . . . 6  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y  =  A  /\  x  e.  y ) )
1514exbii 1635 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  E. y ( y  =  A  /\  x  e.  y ) )
16 vex 3074 . . . . . 6  |-  x  e. 
_V
1716, 1brco 5111 . . . . 5  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. ) )
184clel3 3198 . . . . 5  |-  ( x  e.  A  <->  E. y
( y  =  A  /\  x  e.  y ) )
1915, 17, 183bitr4i 277 . . . 4  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <-> 
x  e.  A )
2010, 1brcnv 5123 . . . . . . . 8  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
214, 5, 10br2ndeq 27723 . . . . . . . 8  |-  ( <. A ,  B >. 2nd y  <->  y  =  B )
2220, 21bitri 249 . . . . . . 7  |-  ( y `' 2nd <. A ,  B >.  <-> 
y  =  B )
239, 22anbi12ci 698 . . . . . 6  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y  =  B  /\  x  e.  y ) )
2423exbii 1635 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  e.  y ) )
2516, 1brco 5111 . . . . 5  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. ) )
265clel3 3198 . . . . 5  |-  ( x  e.  B  <->  E. y
( y  =  B  /\  x  e.  y ) )
2724, 25, 263bitr4i 277 . . . 4  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <-> 
x  e.  B )
2819, 27anbi12i 697 . . 3  |-  ( ( x ( `' 1st  o.  _E  ) <. A ,  B >.  /\  x ( `' 2nd  o.  _E  ) <. A ,  B >. )  <-> 
( x  e.  A  /\  x  e.  B
) )
29 brin 4442 . . 3  |-  ( x ( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  )
) <. A ,  B >.  <-> 
( x ( `' 1st  o.  _E  ) <. A ,  B >.  /\  x ( `' 2nd  o.  _E  ) <. A ,  B >. ) )
30 elin 3640 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3128, 29, 303bitr4ri 278 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x ( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  ) ) <. A ,  B >. )
321, 2, 3, 8, 31brtxpsd3 28064 1  |-  ( <. A ,  B >.Cap C  <-> 
C  =  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3071    i^i cin 3428   <.cop 3984   class class class wbr 4393    _E cep 4731    X. cxp 4939   `'ccnv 4940    o. ccom 4945   1stc1st 6678   2ndc2nd 6679  Capccap 28014
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 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-eprel 4733  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fo 5525  df-fv 5527  df-1st 6680  df-2nd 6681  df-symdif 27986  df-txp 28021  df-cap 28037
This theorem is referenced by:  brrestrict  28117
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