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Theorem sxuni 30061
Description: The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxuni ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))

Proof of Theorem sxuni
StepHypRef Expression
1 sxsigon 30060 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))
2 issgon 29991 . . 3 ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)) ↔ ((𝑆 ×s 𝑇) ∈ ran sigAlgebra ∧ ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇)))
32simprbi 480 . 2 ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
41, 3syl 17 1 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   cuni 4407   × cxp 5077  ran crn 5080  cfv 5852  (class class class)co 6610  sigAlgebracsiga 29975   ×s csx 30056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-siga 29976  df-sigagen 30007  df-sx 30057
This theorem is referenced by:  1stmbfm  30127  2ndmbfm  30128  mbfmco2  30132
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