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Theorem brcap 29167
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 4711 . 2  |-  <. A ,  B >.  e.  _V
2 brcap.3 . 2  |-  C  e. 
_V
3 df-cap 29096 . 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 5045 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 5029 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 918 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 epel 4794 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
10 vex 3116 . . . . . . . . 9  |-  y  e. 
_V
1110, 1brcnv 5183 . . . . . . . 8  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
124, 5, 10br1steq 28781 . . . . . . . 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 1644 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  E. y ( y  =  A  /\  x  e.  y ) )
16 vex 3116 . . . . . 6  |-  x  e. 
_V
1716, 1brco 5171 . . . . 5  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. ) )
184clel3 3242 . . . . 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 5183 . . . . . . . 8  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
214, 5, 10br2ndeq 28782 . . . . . . . 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 1644 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  e.  y ) )
2516, 1brco 5171 . . . . 5  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. ) )
265clel3 3242 . . . . 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 4496 . . 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 3687 . . 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 29123 1  |-  ( <. A ,  B >.Cap C  <-> 
C  =  ( A  i^i  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    i^i cin 3475   <.cop 4033   class class class wbr 4447    _E cep 4789    X. cxp 4997   `'ccnv 4998    o. ccom 5003   1stc1st 6779   2ndc2nd 6780  Capccap 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-eprel 4791  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594  df-1st 6781  df-2nd 6782  df-symdif 29045  df-txp 29080  df-cap 29096
This theorem is referenced by:  brrestrict  29176
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