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Theorem pprodss4v 30209
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 30179 . 2  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
2 txprel 30204 . . 3  |-  Rel  (
( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
3 txpss3v 30203 . . . . . . 7  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( _V  X.  ( _V  X.  _V )
)
43sseli 3437 . . . . . 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 4856 . . . . . 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 4881 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
8 opeq2 4159 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  <. x ,  y
>.  =  <. x , 
<. z ,  w >. >.
)
98eleq1d 2471 . . . . . . . 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 4395 . . . . . . . . 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 3061 . . . . . . . . . . 11  |-  x  e. 
_V
12 vex 3061 . . . . . . . . . . 11  |-  z  e. 
_V
13 vex 3061 . . . . . . . . . . 11  |-  w  e. 
_V
1411, 12, 13brtxp 30205 . . . . . . . . . 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 4993 . . . . . . . . . . . 12  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  <->  E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z ) )
16 vex 3061 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
1716brres 5099 . . . . . . . . . . . . . . 15  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  <->  ( x 1st y  /\  x  e.  ( _V  X.  _V ) ) )
1817simprbi 462 . . . . . . . . . . . . . 14  |-  ( x ( 1st  |`  ( _V  X.  _V ) ) y  ->  x  e.  ( _V  X.  _V )
)
1918adantr 463 . . . . . . . . . . . . 13  |-  ( ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V ) )
2019exlimiv 1743 . . . . . . . . . . . 12  |-  ( E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z )  ->  x  e.  ( _V  X.  _V )
)
2115, 20sylbi 195 . . . . . . . . . . 11  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  ->  x  e.  ( _V  X.  _V )
)
2221adantr 463 . . . . . . . . . 10  |-  ( ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  /\  x ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) w )  ->  x  e.  ( _V  X.  _V )
)
2314, 22sylbi 195 . . . . . . . . 9  |-  ( x ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) <. z ,  w >.  ->  x  e.  ( _V  X.  _V )
)
2410, 23sylbir 213 . . . . . . . 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 228 . . . . . . 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 1744 . . . . . 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 195 . . . . 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 4852 . . . 4  |-  ( <.
x ,  y >.  e.  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )  <-> 
( x  e.  ( _V  X.  _V )  /\  y  e.  ( _V  X.  _V ) ) )
3028, 6, 29sylanbrc 662 . . 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 4914 . 2  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
321, 31eqsstri 3471 1  |- pprod ( A ,  B )  C_  ( ( _V  X.  _V )  X.  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    C_ wss 3413   <.cop 3977   class class class wbr 4394    X. cxp 4820    |` cres 4824    o. ccom 4826   1stc1st 6781   2ndc2nd 6782    (x) ctxp 30154  pprodcpprod 30155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-1st 6783  df-2nd 6784  df-txp 30178  df-pprod 30179
This theorem is referenced by:  brpprod3a  30211
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