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Theorem brpprod 24425
Description: Characterize a quatary relationship over a tail cross product. Together with pprodss4v 24424, 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 24396 . . 3  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
21breqi 4029 . 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 4237 . . 3  |-  <. X ,  Y >.  e.  _V
4 brpprod.3 . . 3  |-  Z  e. 
_V
5 brpprod.4 . . 3  |-  W  e. 
_V
63, 4, 5brtxp 24420 . 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 4852 . . . 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 4735 . . . . . . . 8  |-  <. X ,  Y >.  e.  ( _V 
X.  _V )
11 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
1211brres 4961 . . . . . . . 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 24130 . . . . . . 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 1569 . . . 4  |-  ( E. x ( <. X ,  Y >. ( 1st  |`  ( _V  X.  _V ) ) x  /\  x A Z )  <->  E. x
( x  =  X  /\  x A Z ) )
18 breq1 4026 . . . . 5  |-  ( x  =  X  ->  (
x A Z  <->  X A Z ) )
198, 18ceqsexv 2823 . . . 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 4852 . . . 4  |-  ( <. X ,  Y >. ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) W  <->  E. y ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  /\  y B W ) )
22 vex 2791 . . . . . . . . 9  |-  y  e. 
_V
2322brres 4961 . . . . . . . 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 24131 . . . . . . 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 1569 . . . 4  |-  ( E. y ( <. X ,  Y >. ( 2nd  |`  ( _V  X.  _V ) ) y  /\  y B W )  <->  E. y
( y  =  Y  /\  y B W ) )
29 breq1 4026 . . . . 5  |-  ( y  =  Y  ->  (
y B W  <->  Y B W ) )
309, 29ceqsexv 2823 . . . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    X. cxp 4687    |` cres 4691    o. ccom 4693   1stc1st 6120   2ndc2nd 6121    (x) ctxp 24373  pprodcpprod 24374
This theorem is referenced by:  brpprod3a  24426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-txp 24395  df-pprod 24396
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