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Theorem sxval 26609
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 2986 . . 3  |-  ( S  e.  V  ->  S  e.  _V )
2 elex 2986 . . 3  |-  ( T  e.  W  ->  T  e.  _V )
3 id 22 . . . . . . 7  |-  ( s  =  S  ->  s  =  S )
4 eqidd 2444 . . . . . . 7  |-  ( s  =  S  ->  t  =  t )
5 eqidd 2444 . . . . . . 7  |-  ( s  =  S  ->  (
x  X.  y )  =  ( x  X.  y ) )
63, 4, 5mpt2eq123dv 6153 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  s ,  y  e.  t  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y ) ) )
76rneqd 5072 . . . . 5  |-  ( s  =  S  ->  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y
) )  =  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) ) )
87fveq2d 5700 . . . 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 2444 . . . . . . 7  |-  ( t  =  T  ->  S  =  S )
10 id 22 . . . . . . 7  |-  ( t  =  T  ->  t  =  T )
11 eqidd 2444 . . . . . . 7  |-  ( t  =  T  ->  (
x  X.  y )  =  ( x  X.  y ) )
129, 10, 11mpt2eq123dv 6153 . . . . . 6  |-  ( t  =  T  ->  (
x  e.  S , 
y  e.  t  |->  ( x  X.  y ) )  =  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y ) ) )
1312rneqd 5072 . . . . 5  |-  ( t  =  T  ->  ran  ( x  e.  S ,  y  e.  t  |->  ( x  X.  y
) )  =  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
1413fveq2d 5700 . . . 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 26608 . . . 4  |- ×s  =  ( s  e.  _V ,  t  e. 
_V  |->  (sigaGen `  ran  ( x  e.  s ,  y  e.  t  |->  ( x  X.  y ) ) ) )
16 fvex 5706 . . . 4  |-  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )  e. 
_V
178, 14, 15, 16ovmpt2 6231 . . 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 5699 . 2  |-  (sigaGen `  A
)  =  (sigaGen `  ran  ( x  e.  S ,  y  e.  T  |->  ( x  X.  y
) ) )
2118, 20syl6eqr 2493 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 1369    e. wcel 1756   _Vcvv 2977    X. cxp 4843   ran crn 4846   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098  sigaGencsigagen 26586   ×s csx 26607
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-sx 26608
This theorem is referenced by:  sxsiga  26610  sxsigon  26611  elsx  26613  mbfmco2  26685  sxbrsigalem5  26708  sxbrsiga  26710
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