Theorem sgon

Description: A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017)

	Ref	Expression
Assertion	sgon	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝑆 = ∪ 𝑆
2		issgon	⊢ ( 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) ↔ ( 𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆 ) )
3	2	biimpri	⊢ ( ( 𝑆 ∈ ∪ ran sigAlgebra ∧ ∪ 𝑆 = ∪ 𝑆 ) → 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) )
4	1 3	mpan2	⊢ ( 𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ ( sigAlgebra ‘ ∪ 𝑆 ) )