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Theorem pprodss4v 25729
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 25699 . 2  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
2 txprel 25724 . . 3  |-  Rel  (
( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
3 txpss3v 25723 . . . . . . 7  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( _V  X.  ( _V  X.  _V )
)
43sseli 3344 . . . . . 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 4912 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  ->  y  e.  ( _V  X.  _V )
)
64, 5syl 16 . . . . 5  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  y  e.  ( _V  X.  _V )
)
7 elvv 4936 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
8 opeq2 3985 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  <. x ,  y
>.  =  <. x , 
<. z ,  w >. >.
)
98eleq1d 2502 . . . . . . . 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 4213 . . . . . . . . 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 2959 . . . . . . . . . . 11  |-  x  e. 
_V
12 vex 2959 . . . . . . . . . . 11  |-  z  e. 
_V
13 vex 2959 . . . . . . . . . . 11  |-  w  e. 
_V
1411, 12, 13brtxp 25725 . . . . . . . . . 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 5043 . . . . . . . . . . . 12  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  <->  E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z ) )
16 vex 2959 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
1716brres 5152 . . . . . . . . . . . . . . 15  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  <->  ( x 1st y  /\  x  e.  ( _V  X.  _V ) ) )
1817simprbi 451 . . . . . . . . . . . . . 14  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  ->  x  e.  ( _V  X.  _V )
)
1918adantr 452 . . . . . . . . . . . . 13  |-  ( ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V ) )
2019exlimiv 1644 . . . . . . . . . . . 12  |-  ( E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V )
)
2115, 20sylbi 188 . . . . . . . . . . 11  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  ->  x  e.  ( _V  X.  _V )
)
2221adantr 452 . . . . . . . . . 10  |-  ( ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  /\  x ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) w )  ->  x  e.  ( _V  X.  _V )
)
2314, 22sylbi 188 . . . . . . . . 9  |-  ( x ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) <. z ,  w >.  ->  x  e.  ( _V  X.  _V )
)
2410, 23sylbir 205 . . . . . . . 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 220 . . . . . . 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 1645 . . . . . 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 188 . . . . 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 34 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
)
29 opelxp 4908 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )  <-> 
( x  e.  ( _V  X.  _V )  /\  y  e.  ( _V  X.  _V ) ) )
3028, 6, 29sylanbrc 646 . . 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 4967 . 2  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
321, 31eqsstri 3378 1  |- pprod ( A ,  B )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   <.cop 3817   class class class wbr 4212    X. cxp 4876    |` cres 4880    o. ccom 4882   1stc1st 6347   2ndc2nd 6348    (x) ctxp 25674  pprodcpprod 25675
This theorem is referenced by:  brpprod3a  25731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-txp 25698  df-pprod 25699
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