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Theorem elsx 26606
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 2441 . . . . . 6  |-  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )  =  ran  (
x  e.  S , 
y  e.  T  |->  ( x  X.  y ) )
21txbasex 19137 . . . . 5  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  e. 
_V )
3 sssigagen 26586 . . . . 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 2441 . . . . . 6  |-  ( A  X.  B )  =  ( A  X.  B
)
7 xpeq1 4852 . . . . . . . 8  |-  ( x  =  A  ->  (
x  X.  y )  =  ( A  X.  y ) )
87eqeq2d 2452 . . . . . . 7  |-  ( x  =  A  ->  (
( A  X.  B
)  =  ( x  X.  y )  <->  ( A  X.  B )  =  ( A  X.  y ) ) )
9 xpeq2 4853 . . . . . . . 8  |-  ( y  =  B  ->  ( A  X.  y )  =  ( A  X.  B
) )
109eqeq2d 2452 . . . . . . 7  |-  ( y  =  B  ->  (
( A  X.  B
)  =  ( A  X.  y )  <->  ( A  X.  B )  =  ( A  X.  B ) ) )
118, 10rspc2ev 3079 . . . . . 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 1303 . . . . 5  |-  ( ( A  e.  S  /\  B  e.  T )  ->  E. x  e.  S  E. y  e.  T  ( A  X.  B
)  =  ( x  X.  y ) )
13 xpexg 6505 . . . . . 6  |-  ( ( A  e.  S  /\  B  e.  T )  ->  ( A  X.  B
)  e.  _V )
14 eqid 2441 . . . . . . 7  |-  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )
1514elrnmpt2g 6200 . . . . . 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 3355 . 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 26602 . . 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 2518 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 1369    e. wcel 1756   E.wrex 2714   _Vcvv 2970    C_ wss 3326    X. cxp 4836   ran crn 4839   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091  sigaGencsigagen 26579   ×s csx 26600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-siga 26549  df-sigagen 26580  df-sx 26601
This theorem is referenced by:  1stmbfm  26673  2ndmbfm  26674
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