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Theorem sxval 27829
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 3122 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 3122 . . 3  |-  ( T  e.  W  ->  T  e.  _V )
3 id 22 . . . . . . 7  |-  ( s  =  S  ->  s  =  S )
4 eqidd 2468 . . . . . . 7  |-  ( s  =  S  ->  t  =  t )
5 eqidd 2468 . . . . . . 7  |-  ( s  =  S  ->  (
x  X.  y )  =  ( x  X.  y ) )
63, 4, 5mpt2eq123dv 6343 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  s ,  y  e.  t  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y ) ) )
76rneqd 5230 . . . . 5  |-  ( s  =  S  ->  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y
) )  =  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) ) )
87fveq2d 5870 . . . 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 2468 . . . . . . 7  |-  ( t  =  T  ->  S  =  S )
10 id 22 . . . . . . 7  |-  ( t  =  T  ->  t  =  T )
11 eqidd 2468 . . . . . . 7  |-  ( t  =  T  ->  (
x  X.  y )  =  ( x  X.  y ) )
129, 10, 11mpt2eq123dv 6343 . . . . . 6  |-  ( t  =  T  ->  (
x  e.  S , 
y  e.  t  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) )
1312rneqd 5230 . . . . 5  |-  ( t  =  T  ->  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) )  =  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
1413fveq2d 5870 . . . 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 27828 . . . 4  |- ×s  =  ( s  e.  _V ,  t  e. 
_V  |->  (sigaGen `  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y ) ) ) )
16 fvex 5876 . . . 4  |-  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )  e. 
_V
178, 14, 15, 16ovmpt2 6422 . . 3  |-  ( ( S  e.  _V  /\  T  e.  _V )  ->  ( S ×s  T )  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) ) )
181, 2, 17syl2an 477 . 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 5869 . 2  |-  (sigaGen `  A
)  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
2118, 20syl6eqr 2526 1  |-  ( ( S  e.  V  /\  T  e.  W )  ->  ( S ×s  T )  =  (sigaGen `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-sx 27828
This theorem is referenced by:  sxsiga  27830  sxsigon  27831  elsx  27833  mbfmco2  27904  sxbrsigalem5  27927  sxbrsiga  27929
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