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Theorem elsx 29030
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 2423 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
21txbasex 20585 . . . . 5  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  e. 
_V )
3 sssigagen 28981 . . . . 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 17 . . . 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 467 . . 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 2423 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4873 . . . . . . . 8  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
87eqeq2d 2437 . . . . . . 7  |-  ( x  =  A  ->  (
( A  X.  B
)  =  ( x  X.  y )  <->  ( A  X.  B )  =  ( A  X.  y ) ) )
9 xpeq2 4874 . . . . . . . 8  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
109eqeq2d 2437 . . . . . . 7  |-  ( y  =  B  ->  (
( A  X.  B
)  =  ( A  X.  y )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 3199 . . . . . 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 1350 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  T )  ->  E. x  e.  S  E. y  e.  T  ( A  X.  B
)  =  ( x  X.  y ) )
13 xpexg 6613 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2423 . . . . . . 7  |-  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )
1514elrnmpt2g 6428 . . . . . 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 17 . . . . 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 236 . . . 4  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) ) )
1817adantl 468 . . 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 3471 . 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 29026 . . 3  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
2120adantr 467 . 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 2515 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 188    /\ wa 371    = wceq 1438    e. wcel 1873   E.wrex 2777   _Vcvv 3085    C_ wss 3442    X. cxp 4857   ran crn 4860   ` cfv 5607  (class class class)co 6311    |-> cmpt2 6313  sigaGencsigagen 28974   ×s csx 29024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-id 4774  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-fv 5615  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-1st 6813  df-2nd 6814  df-siga 28944  df-sigagen 28975  df-sx 29025
This theorem is referenced by:  1stmbfm  29096  2ndmbfm  29097
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