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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
Assertion
Ref Expression
sxval ×s sigaGen
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem sxval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3096 . . 3
2 elex 3096 . . 3
3 id 23 . . . . . . 7
4 eqidd 2430 . . . . . . 7
5 eqidd 2430 . . . . . . 7
63, 4, 5mpt2eq123dv 6367 . . . . . 6
76rneqd 5082 . . . . 5
87fveq2d 5885 . . . 4 sigaGen sigaGen
9 eqidd 2430 . . . . . . 7
10 id 23 . . . . . . 7
11 eqidd 2430 . . . . . . 7
129, 10, 11mpt2eq123dv 6367 . . . . . 6
1312rneqd 5082 . . . . 5
1413fveq2d 5885 . . . 4 sigaGen sigaGen
15 df-sx 28850 . . . 4 ×s sigaGen
16 fvex 5891 . . . 4 sigaGen
178, 14, 15, 16ovmpt2 6446 . . 3 ×s sigaGen
181, 2, 17syl2an 479 . 2 ×s sigaGen
19 sxval.1 . . 3
2019fveq2i 5884 . 2 sigaGen sigaGen
2118, 20syl6eqr 2488 1 ×s sigaGen
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1870  cvv 3087   cxp 4852   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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