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Theorem brpprod 24496
Description: Characterize a quatary relationship over a tail cross product. Together with pprodss4v 24495, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brpprod.1  |-  X  e. 
_V
brpprod.2  |-  Y  e. 
_V
brpprod.3  |-  Z  e. 
_V
brpprod.4  |-  W  e. 
_V
Assertion
Ref Expression
brpprod  |-  ( <. X ,  Y >.pprod ( A ,  B )
<. Z ,  W >.  <->  ( X A Z  /\  Y B W ) )

Proof of Theorem brpprod
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pprod 24467 . . 3  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
21breqi 4045 . 2  |-  ( <. X ,  Y >.pprod ( A ,  B )
<. Z ,  W >.  <->  <. X ,  Y >. (
( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
<. Z ,  W >. )
3 opex 4253 . . 3  |-  <. X ,  Y >.  e.  _V
4 brpprod.3 . . 3  |-  Z  e. 
_V
5 brpprod.4 . . 3  |-  W  e. 
_V
63, 4, 5brtxp 24491 . 2  |-  ( <. X ,  Y >. ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
<. Z ,  W >.  <->  ( <. X ,  Y >. ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) Z  /\  <. X ,  Y >. ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) W ) )
73, 4brco 4868 . . . 4  |-  ( <. X ,  Y >. ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) Z  <->  E. x ( <. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  /\  x A Z ) )
8 brpprod.1 . . . . . . . . 9  |-  X  e. 
_V
9 brpprod.2 . . . . . . . . 9  |-  Y  e. 
_V
108, 9opelvv 4751 . . . . . . . 8  |-  <. X ,  Y >.  e.  ( _V 
X.  _V )
11 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
1211brres 4977 . . . . . . . 8  |-  ( <. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  <-> 
( <. X ,  Y >. 1st x  /\  <. X ,  Y >.  e.  ( _V  X.  _V )
) )
1310, 12mpbiran2 885 . . . . . . 7  |-  ( <. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  <->  <. X ,  Y >. 1st x )
148, 9, 11br1steq 24201 . . . . . . 7  |-  ( <. X ,  Y >. 1st x  <->  x  =  X
)
1513, 14bitri 240 . . . . . 6  |-  ( <. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  <-> 
x  =  X )
1615anbi1i 676 . . . . 5  |-  ( (
<. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  /\  x A Z )  <->  ( x  =  X  /\  x A Z ) )
1716exbii 1572 . . . 4  |-  ( E. x ( <. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  /\  x A Z )  <->  E. x
( x  =  X  /\  x A Z ) )
18 breq1 4042 . . . . 5  |-  ( x  =  X  ->  (
x A Z  <->  X A Z ) )
198, 18ceqsexv 2836 . . . 4  |-  ( E. x ( x  =  X  /\  x A Z )  <->  X A Z )
207, 17, 193bitri 262 . . 3  |-  ( <. X ,  Y >. ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) Z  <-> 
X A Z )
213, 5brco 4868 . . . 4  |-  ( <. X ,  Y >. ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) W  <->  E. y ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  /\  y B W ) )
22 vex 2804 . . . . . . . . 9  |-  y  e. 
_V
2322brres 4977 . . . . . . . 8  |-  ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  <-> 
( <. X ,  Y >. 2nd y  /\  <. X ,  Y >.  e.  ( _V  X.  _V )
) )
2410, 23mpbiran2 885 . . . . . . 7  |-  ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  <->  <. X ,  Y >. 2nd y )
258, 9, 22br2ndeq 24202 . . . . . . 7  |-  ( <. X ,  Y >. 2nd y  <->  y  =  Y )
2624, 25bitri 240 . . . . . 6  |-  ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  <-> 
y  =  Y )
2726anbi1i 676 . . . . 5  |-  ( (
<. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  /\  y B W )  <->  ( y  =  Y  /\  y B W ) )
2827exbii 1572 . . . 4  |-  ( E. y ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  /\  y B W )  <->  E. y
( y  =  Y  /\  y B W ) )
29 breq1 4042 . . . . 5  |-  ( y  =  Y  ->  (
y B W  <->  Y B W ) )
309, 29ceqsexv 2836 . . . 4  |-  ( E. y ( y  =  Y  /\  y B W )  <->  Y B W )
3121, 28, 303bitri 262 . . 3  |-  ( <. X ,  Y >. ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) W  <-> 
Y B W )
3220, 31anbi12i 678 . 2  |-  ( (
<. X ,  Y >. ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) Z  /\  <. X ,  Y >. ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) W )  <->  ( X A Z  /\  Y B W ) )
332, 6, 323bitri 262 1  |-  ( <. X ,  Y >.pprod ( A ,  B )
<. Z ,  W >.  <->  ( X A Z  /\  Y B W ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703    |` cres 4707    o. ccom 4709   1stc1st 6136   2ndc2nd 6137    (x) ctxp 24444  pprodcpprod 24445
This theorem is referenced by:  brpprod3a  24497
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-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-txp 24466  df-pprod 24467
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