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Theorem sxval 28851
Description: Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Hypothesis
Ref Expression
sxval.1  |-  A  =  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )
Assertion
Ref Expression
sxval  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  A ) )
Distinct variable groups:    x, y, S    x, T, y
Allowed substitution hints:    A( x, y)    V( x, y)    W( x, y)

Proof of Theorem sxval
Dummy variables  t 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3096 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 3096 . . 3  |-  ( T  e.  W  ->  T  e.  _V )
3 id 23 . . . . . . 7  |-  ( s  =  S  ->  s  =  S )
4 eqidd 2430 . . . . . . 7  |-  ( s  =  S  ->  t  =  t )
5 eqidd 2430 . . . . . . 7  |-  ( s  =  S  ->  (
x  X.  y )  =  ( x  X.  y ) )
63, 4, 5mpt2eq123dv 6367 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  s ,  y  e.  t  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y ) ) )
76rneqd 5082 . . . . 5  |-  ( s  =  S  ->  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y
) )  =  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) ) )
87fveq2d 5885 . . . 4  |-  ( s  =  S  ->  (sigaGen ` 
ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y ) ) )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) ) ) )
9 eqidd 2430 . . . . . . 7  |-  ( t  =  T  ->  S  =  S )
10 id 23 . . . . . . 7  |-  ( t  =  T  ->  t  =  T )
11 eqidd 2430 . . . . . . 7  |-  ( t  =  T  ->  (
x  X.  y )  =  ( x  X.  y ) )
129, 10, 11mpt2eq123dv 6367 . . . . . 6  |-  ( t  =  T  ->  (
x  e.  S , 
y  e.  t  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) )
1312rneqd 5082 . . . . 5  |-  ( t  =  T  ->  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) )  =  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
1413fveq2d 5885 . . . 4  |-  ( t  =  T  ->  (sigaGen ` 
ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y ) ) )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) ) )
15 df-sx 28850 . . . 4  |- ×s  =  ( s  e.  _V ,  t  e. 
_V  |->  (sigaGen `  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y ) ) ) )
16 fvex 5891 . . . 4  |-  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )  e. 
_V
178, 14, 15, 16ovmpt2 6446 . . 3  |-  ( ( S  e.  _V  /\  T  e.  _V )  ->  ( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
181, 2, 17syl2an 479 . 2  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
19 sxval.1 . . 3  |-  A  =  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) )
2019fveq2i 5884 . 2  |-  (sigaGen `  A
)  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
2118, 20syl6eqr 2488 1  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  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-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-sx 28850
This theorem is referenced by:  sxsiga  28852  sxsigon  28853  elsx  28855  mbfmco2  28926  sxbrsigalem5  28949  sxbrsiga  28951
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