Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcup Unicode version

Theorem brcup 24549
Description: Binary relationship form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcup.1  |-  A  e. 
_V
brcup.2  |-  B  e. 
_V
brcup.3  |-  C  e. 
_V
Assertion
Ref Expression
brcup  |-  ( <. A ,  B >.Cup C  <-> 
C  =  ( A  u.  B ) )

Proof of Theorem brcup
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4253 . 2  |-  <. A ,  B >.  e.  _V
2 brcup.3 . 2  |-  C  e. 
_V
3 df-cup 24481 . 2  |- Cup  =  ( ( ( _V  X.  _V )  X.  _V )  \  ran  ( ( _V 
(x)  _E  )(++) (
( ( `' 1st  o.  _E  )  u.  ( `' 2nd  o.  _E  )
)  (x)  _V )
) )
4 brcup.1 . . . 4  |-  A  e. 
_V
5 brcup.2 . . . 4  |-  B  e. 
_V
64, 5opelvv 4751 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4736 . . 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 2804 . . . . . . 7  |-  x  e. 
_V
109, 1brco 4868 . . . . . 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 2804 . . . . . . . . . . 11  |-  y  e. 
_V
1312, 1brcnv 4880 . . . . . . . . . 10  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
144, 5, 12br1steq 24201 . . . . . . . . . 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 1572 . . . . . 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 4043 . . . . . 6  |-  ( y  =  A  ->  (
x  _E  y  <->  x  _E  A ) )
214, 20ceqsexv 2836 . . . . 5  |-  ( E. y ( y  =  A  /\  x  _E  y )  <->  x  _E  A )
224epelc 4323 . . . . 5  |-  ( x  _E  A  <->  x  e.  A )
2319, 21, 223bitri 262 . . . 4  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <-> 
x  e.  A )
249, 1brco 4868 . . . . 5  |-  ( 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 4880 . . . . . . . . . 10  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
274, 5, 12br2ndeq 24202 . . . . . . . . . 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 1572 . . . . . 6  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  _E  y ) )
32 breq2 4043 . . . . . . 7  |-  ( y  =  B  ->  (
x  _E  y  <->  x  _E  B ) )
335, 32ceqsexv 2836 . . . . . 6  |-  ( E. y ( y  =  B  /\  x  _E  y )  <->  x  _E  B )
3431, 33bitri 240 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  x  _E  B
)
355epelc 4323 . . . . 5  |-  ( x  _E  B  <->  x  e.  B )
3624, 34, 353bitri 262 . . . 4  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <-> 
x  e.  B )
3723, 36orbi12i 507 . . 3  |-  ( ( x ( `' 1st  o.  _E  ) <. A ,  B >.  \/  x ( `' 2nd  o.  _E  ) <. A ,  B >. )  <-> 
( x  e.  A  \/  x  e.  B
) )
38 brun 4085 . . 3  |-  ( x ( ( `' 1st  o.  _E  )  u.  ( `' 2nd  o.  _E  )
) <. A ,  B >.  <-> 
( x ( `' 1st  o.  _E  ) <. A ,  B >.  \/  x ( `' 2nd  o.  _E  ) <. A ,  B >. ) )
39 elun 3329 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
4037, 38, 393bitr4ri 269 . 2  |-  ( x  e.  ( A  u.  B )  <->  x (
( `' 1st  o.  _E  )  u.  ( `' 2nd  o.  _E  )
) <. A ,  B >. )
411, 2, 3, 8, 40brtxpsd3 24507 1  |-  ( <. A ,  B >.Cup C  <-> 
C  =  ( A  u.  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   <.cop 3656   class class class wbr 4039    _E cep 4319    X. cxp 4703   `'ccnv 4704    o. ccom 4709   1stc1st 6136   2ndc2nd 6137  Cupccup 24460
This theorem is referenced by:  brsuccf  24551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-cup 24481
  Copyright terms: Public domain W3C validator