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Theorem brcap 25777
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 4419 . 2  |-  <. A ,  B >.  e.  _V
2 brcap.3 . 2  |-  C  e. 
_V
3 df-cap 25706 . 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 4916 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4901 . . 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 epel 4489 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
10 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
1110, 1brcnv 5047 . . . . . . . 8  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
124, 5, 10br1steq 25390 . . . . . . . 8  |-  ( <. A ,  B >. 1st y  <->  y  =  A )
1311, 12bitri 241 . . . . . . 7  |-  ( y `' 1st <. A ,  B >.  <-> 
y  =  A )
149, 13anbi12ci 680 . . . . . 6  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y  =  A  /\  x  e.  y ) )
1514exbii 1592 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  E. y ( y  =  A  /\  x  e.  y ) )
16 vex 2951 . . . . . 6  |-  x  e. 
_V
1716, 1brco 5035 . . . . 5  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. ) )
184clel3 3066 . . . . 5  |-  ( x  e.  A  <->  E. y
( y  =  A  /\  x  e.  y ) )
1915, 17, 183bitr4i 269 . . . 4  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <-> 
x  e.  A )
2010, 1brcnv 5047 . . . . . . . 8  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
214, 5, 10br2ndeq 25391 . . . . . . . 8  |-  ( <. A ,  B >. 2nd y  <->  y  =  B )
2220, 21bitri 241 . . . . . . 7  |-  ( y `' 2nd <. A ,  B >.  <-> 
y  =  B )
239, 22anbi12ci 680 . . . . . 6  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y  =  B  /\  x  e.  y ) )
2423exbii 1592 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  e.  y ) )
2516, 1brco 5035 . . . . 5  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. ) )
265clel3 3066 . . . . 5  |-  ( x  e.  B  <->  E. y
( y  =  B  /\  x  e.  y ) )
2724, 25, 263bitr4i 269 . . . 4  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <-> 
x  e.  B )
2819, 27anbi12i 679 . . 3  |-  ( ( x ( `' 1st  o.  _E  ) <. A ,  B >.  /\  x ( `' 2nd  o.  _E  ) <. A ,  B >. )  <-> 
( x  e.  A  /\  x  e.  B
) )
29 brin 4251 . . 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 3522 . . 3  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3128, 29, 303bitr4ri 270 . 2  |-  ( x  e.  ( A  i^i  B )  <->  x ( ( `' 1st  o.  _E  )  i^i  ( `' 2nd  o.  _E  ) ) <. A ,  B >. )
321, 2, 3, 8, 31brtxpsd3 25733 1  |-  ( <. A ,  B >.Cap C  <-> 
C  =  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   <.cop 3809   class class class wbr 4204    _E cep 4484    X. cxp 4868   `'ccnv 4869    o. ccom 4874   1stc1st 6339   2ndc2nd 6340  Capccap 25683
This theorem is referenced by:  brrestrict  25786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-eprel 4486  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-1st 6341  df-2nd 6342  df-symdif 25655  df-txp 25690  df-cap 25706
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