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Theorem elsx 27833
Description: The cartesian product of two open sets is an element of the product sigma algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
Assertion
Ref Expression
elsx  |-  ( ( ( S  e.  V  /\  T  e.  W
)  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A  X.  B
)  e.  ( S ×s  T ) )

Proof of Theorem elsx
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
21txbasex 19830 . . . . 5  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  e. 
_V )
3 sssigagen 27813 . . . . 5  |-  ( ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  e.  _V  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  C_  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
42, 3syl 16 . . . 4  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  C_  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
54adantr 465 . . 3  |-  ( ( ( S  e.  V  /\  T  e.  W
)  /\  ( A  e.  S  /\  B  e.  T ) )  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  C_  (sigaGen ` 
ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
6 eqid 2467 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 5013 . . . . . . . 8  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
87eqeq2d 2481 . . . . . . 7  |-  ( x  =  A  ->  (
( A  X.  B
)  =  ( x  X.  y )  <->  ( A  X.  B )  =  ( A  X.  y ) ) )
9 xpeq2 5014 . . . . . . . 8  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
109eqeq2d 2481 . . . . . . 7  |-  ( y  =  B  ->  (
( A  X.  B
)  =  ( A  X.  y )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 3225 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  T  /\  ( A  X.  B
)  =  ( A  X.  B ) )  ->  E. x  e.  S  E. y  e.  T  ( A  X.  B
)  =  ( x  X.  y ) )
126, 11mp3an3 1313 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  T )  ->  E. x  e.  S  E. y  e.  T  ( A  X.  B
)  =  ( x  X.  y ) )
13 xpexg 6586 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2467 . . . . . . 7  |-  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )
1514elrnmpt2g 6398 . . . . . 6  |-  ( ( A  X.  B )  e.  _V  ->  (
( A  X.  B
)  e.  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  <->  E. x  e.  S  E. y  e.  T  ( A  X.  B
)  =  ( x  X.  y ) ) )
1613, 15syl 16 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( ( A  X.  B )  e.  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) )  <->  E. x  e.  S  E. y  e.  T  ( A  X.  B )  =  ( x  X.  y ) ) )
1712, 16mpbird 232 . . . 4  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) ) )
1817adantl 466 . . 3  |-  ( ( ( S  e.  V  /\  T  e.  W
)  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A  X.  B
)  e.  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) ) )
195, 18sseldd 3505 . 2  |-  ( ( ( S  e.  V  /\  T  e.  W
)  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A  X.  B
)  e.  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) ) )
201sxval 27829 . . 3  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
2120adantr 465 . 2  |-  ( ( ( S  e.  V  /\  T  e.  W
)  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
2219, 21eleqtrrd 2558 1  |-  ( ( ( S  e.  V  /\  T  e.  W
)  /\  ( A  e.  S  /\  B  e.  T ) )  -> 
( A  X.  B
)  e.  ( S ×s  T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476    X. cxp 4997   ran crn 5000   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286  sigaGencsigagen 27806   ×s csx 27827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-siga 27776  df-sigagen 27807  df-sx 27828
This theorem is referenced by:  1stmbfm  27899  2ndmbfm  27900
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