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Theorem pprodss4v 30651
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 30621 . 2  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
2 txprel 30646 . . 3  |-  Rel  (
( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
3 txpss3v 30645 . . . . . . 7  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( _V  X.  ( _V  X.  _V )
)
43sseli 3428 . . . . . 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 4868 . . . . . 6  |-  ( <.
x ,  y >.  e.  ( _V  X.  ( _V  X.  _V ) )  ->  y  e.  ( _V  X.  _V )
)
64, 5syl 17 . . . . 5  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  y  e.  ( _V  X.  _V )
)
7 elvv 4893 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
8 opeq2 4167 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  <. x ,  y
>.  =  <. x , 
<. z ,  w >. >.
)
98eleq1d 2513 . . . . . . . 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 4403 . . . . . . . . 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 3048 . . . . . . . . . . 11  |-  x  e. 
_V
12 vex 3048 . . . . . . . . . . 11  |-  z  e. 
_V
13 vex 3048 . . . . . . . . . . 11  |-  w  e. 
_V
1411, 12, 13brtxp 30647 . . . . . . . . . 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 5005 . . . . . . . . . . . 12  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  <->  E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z ) )
16 vex 3048 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
1716brres 5111 . . . . . . . . . . . . . . 15  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  <->  ( x 1st y  /\  x  e.  ( _V  X.  _V ) ) )
1817simprbi 466 . . . . . . . . . . . . . 14  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  ->  x  e.  ( _V  X.  _V )
)
1918adantr 467 . . . . . . . . . . . . 13  |-  ( ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V ) )
2019exlimiv 1776 . . . . . . . . . . . 12  |-  ( E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V )
)
2115, 20sylbi 199 . . . . . . . . . . 11  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  ->  x  e.  ( _V  X.  _V )
)
2221adantr 467 . . . . . . . . . 10  |-  ( ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  /\  x ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) w )  ->  x  e.  ( _V  X.  _V )
)
2314, 22sylbi 199 . . . . . . . . 9  |-  ( x ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) <. z ,  w >.  ->  x  e.  ( _V  X.  _V )
)
2410, 23sylbir 217 . . . . . . . 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 232 . . . . . . 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 1778 . . . . . 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 199 . . . . 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 37 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  ->  x  e.  ( _V  X.  _V )
)
29 opelxp 4864 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )  <-> 
( x  e.  ( _V  X.  _V )  /\  y  e.  ( _V  X.  _V ) ) )
3028, 6, 29sylanbrc 670 . . 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 4926 . 2  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
321, 31eqsstri 3462 1  |- pprod ( A ,  B )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045    C_ wss 3404   <.cop 3974   class class class wbr 4402    X. cxp 4832    |` cres 4836    o. ccom 4838   1stc1st 6791   2ndc2nd 6792    (x) ctxp 30596  pprodcpprod 30597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-1st 6793  df-2nd 6794  df-txp 30620  df-pprod 30621
This theorem is referenced by:  brpprod3a  30653
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