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Theorem brcup 30264
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 4654 . 2  |-  <. A ,  B >.  e.  _V
2 brcup.3 . 2  |-  C  e. 
_V
3 df-cup 30193 . 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 4869 . . 3  |-  <. A ,  B >.  e.  ( _V 
X.  _V )
7 brxp 4853 . . 3  |-  ( <. A ,  B >. ( ( _V  X.  _V )  X.  _V ) C  <-> 
( <. A ,  B >.  e.  ( _V  X.  _V )  /\  C  e. 
_V ) )
86, 2, 7mpbir2an 921 . 2  |-  <. A ,  B >. ( ( _V 
X.  _V )  X.  _V ) C
9 epel 4736 . . . . . . 7  |-  ( x  _E  y  <->  x  e.  y )
10 vex 3061 . . . . . . . . 9  |-  y  e. 
_V
1110, 1brcnv 5005 . . . . . . . 8  |-  ( y `' 1st <. A ,  B >.  <->  <. A ,  B >. 1st y )
124, 5, 10br1steq 29974 . . . . . . . 8  |-  ( <. A ,  B >. 1st y  <->  y  =  A )
1311, 12bitri 249 . . . . . . 7  |-  ( y `' 1st <. A ,  B >.  <-> 
y  =  A )
149, 13anbi12ci 696 . . . . . 6  |-  ( ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  ( y  =  A  /\  x  e.  y ) )
1514exbii 1688 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. )  <->  E. y ( y  =  A  /\  x  e.  y ) )
16 vex 3061 . . . . . 6  |-  x  e. 
_V
1716, 1brco 4993 . . . . 5  |-  ( x ( `' 1st  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 1st <. A ,  B >. ) )
184clel3 3187 . . . . 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 5005 . . . . . . . 8  |-  ( y `' 2nd <. A ,  B >.  <->  <. A ,  B >. 2nd y )
214, 5, 10br2ndeq 29975 . . . . . . . 8  |-  ( <. A ,  B >. 2nd y  <->  y  =  B )
2220, 21bitri 249 . . . . . . 7  |-  ( y `' 2nd <. A ,  B >.  <-> 
y  =  B )
239, 22anbi12ci 696 . . . . . 6  |-  ( ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  ( y  =  B  /\  x  e.  y ) )
2423exbii 1688 . . . . 5  |-  ( E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. )  <->  E. y ( y  =  B  /\  x  e.  y ) )
2516, 1brco 4993 . . . . 5  |-  ( x ( `' 2nd  o.  _E  ) <. A ,  B >.  <->  E. y ( x  _E  y  /\  y `' 2nd <. A ,  B >. ) )
265clel3 3187 . . . . 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, 27orbi12i 519 . . 3  |-  ( ( x ( `' 1st  o.  _E  ) <. A ,  B >.  \/  x ( `' 2nd  o.  _E  ) <. A ,  B >. )  <-> 
( x  e.  A  \/  x  e.  B
) )
29 brun 4442 . . 3  |-  ( x ( ( `' 1st  o.  _E  )  u.  ( `' 2nd  o.  _E  )
) <. A ,  B >.  <-> 
( x ( `' 1st  o.  _E  ) <. A ,  B >.  \/  x ( `' 2nd  o.  _E  ) <. A ,  B >. ) )
30 elun 3583 . . 3  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
3128, 29, 303bitr4ri 278 . 2  |-  ( x  e.  ( A  u.  B )  <->  x (
( `' 1st  o.  _E  )  u.  ( `' 2nd  o.  _E  )
) <. A ,  B >. )
321, 2, 3, 8, 31brtxpsd3 30221 1  |-  ( <. A ,  B >.Cup C  <-> 
C  =  ( A  u.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    u. cun 3411   <.cop 3977   class class class wbr 4394    _E cep 4731    X. cxp 4820   `'ccnv 4821    o. ccom 4826   1stc1st 6781   2ndc2nd 6782  Cupccup 30170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-symdif 3669  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-eprel 4733  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-1st 6783  df-2nd 6784  df-txp 30178  df-cup 30193
This theorem is referenced by:  brsuccf  30266
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