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Theorem pprodss4v 24495
Description: The parallel product is a subclass of  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) ). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
pprodss4v  |- pprod ( A ,  B )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )

Proof of Theorem pprodss4v
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pprod 24467 . 2  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
2 txprel 24490 . . 3  |-  Rel  (
( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
3 txpss3v 24489 . . . . . . 7  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( _V  X.  ( _V  X.  _V )
)
43sseli 3189 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  <. x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) ) )
5 opelxp2 4739 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  ->  y  e.  ( _V  X.  _V )
)
64, 5syl 15 . . . . 5  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  y  e.  ( _V  X.  _V )
)
7 elvv 4764 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
8 opeq2 3813 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  <. x ,  y
>.  =  <. x , 
<. z ,  w >. >.
)
98eleq1d 2362 . . . . . . . 8  |-  ( y  =  <. z ,  w >.  ->  ( <. x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  <->  <. x ,  <. z ,  w >. >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) ) )
10 df-br 4040 . . . . . . . . 9  |-  ( x ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) <. z ,  w >.  <->  <. x ,  <. z ,  w >. >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
11 vex 2804 . . . . . . . . . . 11  |-  x  e. 
_V
12 vex 2804 . . . . . . . . . . 11  |-  z  e. 
_V
13 vex 2804 . . . . . . . . . . 11  |-  w  e. 
_V
1411, 12, 13brtxp 24491 . . . . . . . . . 10  |-  ( x ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) <. z ,  w >.  <-> 
( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  /\  x
( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) w ) )
1511, 12brco 4868 . . . . . . . . . . . 12  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  <->  E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z ) )
16 vex 2804 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
1716brres 4977 . . . . . . . . . . . . . . 15  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  <->  ( x 1st y  /\  x  e.  ( _V  X.  _V ) ) )
1817simprbi 450 . . . . . . . . . . . . . 14  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  ->  x  e.  ( _V  X.  _V )
)
1918adantr 451 . . . . . . . . . . . . 13  |-  ( ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V ) )
2019exlimiv 1624 . . . . . . . . . . . 12  |-  ( E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V )
)
2115, 20sylbi 187 . . . . . . . . . . 11  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  ->  x  e.  ( _V  X.  _V )
)
2221adantr 451 . . . . . . . . . 10  |-  ( ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  /\  x ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) w )  ->  x  e.  ( _V  X.  _V )
)
2314, 22sylbi 187 . . . . . . . . 9  |-  ( x ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) <. z ,  w >.  ->  x  e.  ( _V  X.  _V )
)
2410, 23sylbir 204 . . . . . . . 8  |-  ( <.
x ,  <. z ,  w >. >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
)
259, 24syl6bi 219 . . . . . . 7  |-  ( y  =  <. z ,  w >.  ->  ( <. x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
) )
2625exlimivv 1625 . . . . . 6  |-  ( E. z E. w  y  =  <. z ,  w >.  ->  ( <. x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
) )
277, 26sylbi 187 . . . . 5  |-  ( y  e.  ( _V  X.  _V )  ->  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
) )
286, 27mpcom 32 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
)
29 opelxp 4735 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )  <-> 
( x  e.  ( _V  X.  _V )  /\  y  e.  ( _V  X.  _V ) ) )
3028, 6, 29sylanbrc 645 . . 3  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  <. x ,  y >.  e.  (
( _V  X.  _V )  X.  ( _V  X.  _V ) ) )
312, 30relssi 4794 . 2  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
321, 31eqsstri 3221 1  |- pprod ( A ,  B )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   <.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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