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Theorem pprodss4v 23835
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 23807 . 2  |- pprod ( A ,  B )  =  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) 
(x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
2 txprel 23830 . . 3  |-  Rel  (
( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )
3 txpss3v 23829 . . . . . . 7  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( _V  X.  ( _V  X.  _V )
)
43sseli 3176 . . . . . 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 4723 . . . . . 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 4748 . . . . . 6  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
8 opeq2 3797 . . . . . . . . 9  |-  ( y  =  <. z ,  w >.  ->  <. x ,  y
>.  =  <. x , 
<. z ,  w >. >.
)
98eleq1d 2349 . . . . . . . 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 4024 . . . . . . . . 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 2791 . . . . . . . . . . 11  |-  x  e. 
_V
12 vex 2791 . . . . . . . . . . 11  |-  z  e. 
_V
13 vex 2791 . . . . . . . . . . 11  |-  w  e. 
_V
1411, 12, 13brtxp 23831 . . . . . . . . . 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 4852 . . . . . . . . . . . 12  |-  ( x ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) ) z  <->  E. y ( x ( 1st  |`  ( _V  X.  _V ) ) y  /\  y A z ) )
16 vex 2791 . . . . . . . . . . . . . . . 16  |-  y  e. 
_V
1716brres 4961 . . . . . . . . . . . . . . 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 1666 . . . . . . . . . . . 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 1667 . . . . . 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 4719 . . . 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 4778 . 2  |-  ( ( A  o.  ( 1st  |`  ( _V  X.  _V ) ) )  (x)  ( B  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) 
C_  ( ( _V 
X.  _V )  X.  ( _V  X.  _V ) )
321, 31eqsstri 3208 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643   class class class wbr 4023    X. cxp 4687    |` cres 4691    o. ccom 4693   1stc1st 6120   2ndc2nd 6121    (x) ctxp 23784  pprodcpprod 23785
This theorem is referenced by:  brpprod3a  23837
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 23806  df-pprod 23807
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