Theorem sxuni

Description: The base set of a product sigma-algebra. (Contributed by Thierry Arnoux, 1-Jun-2017)

	Ref	Expression
Assertion	sxuni	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra ) → ( ∪ 𝑆 × ∪ 𝑇 ) = ∪ ( 𝑆 ×s 𝑇 ) )

Proof

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