Theorem sxuni 34356

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

Step	Hyp	Ref	Expression
1		sxsigon 34355	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘(∪ 𝑆 × ∪ 𝑇)))
2		issgon 34286	. . 3 ⊢ ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘(∪ 𝑆 × ∪ 𝑇)) ↔ ((𝑆 ×s 𝑇) ∈ ∪ ran sigAlgebra ∧ (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇)))
3	2	simprbi 497	. 2 ⊢ ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘(∪ 𝑆 × ∪ 𝑇)) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))
4	1, 3	syl 17	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪ (𝑆 ×s 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∪ cuni 4851   × cxp 5623  ran crn 5626  ‘cfv 6493  (class class class)co 7361  sigAlgebracsiga 34271   ×_s csx 34351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-siga 34272  df-sigagen 34302  df-sx 34352

This theorem is referenced by:  1stmbfm  34423  2ndmbfm  34424  mbfmco2  34428