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Theorem elsx 28855
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 2429 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
21txbasex 20512 . . . . 5  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  e. 
_V )
3 sssigagen 28806 . . . . 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 466 . . 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 2429 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4868 . . . . . . . 8  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
87eqeq2d 2443 . . . . . . 7  |-  ( x  =  A  ->  (
( A  X.  B
)  =  ( x  X.  y )  <->  ( A  X.  B )  =  ( A  X.  y ) ) )
9 xpeq2 4869 . . . . . . . 8  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
109eqeq2d 2443 . . . . . . 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 1349 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  T )  ->  E. x  e.  S  E. y  e.  T  ( A  X.  B
)  =  ( x  X.  y ) )
13 xpexg 6607 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2429 . . . . . . 7  |-  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )
1514elrnmpt2g 6422 . . . . . 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 235 . . . 4  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) ) )
1817adantl 467 . . 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 28851 . . 3  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
2120adantr 466 . 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 2520 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   _Vcvv 3087    C_ wss 3442    X. cxp 4852   ran crn 4855   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307  sigaGencsigagen 28799   ×s csx 28849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-siga 28769  df-sigagen 28800  df-sx 28850
This theorem is referenced by:  1stmbfm  28921  2ndmbfm  28922
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