Theorem sssigagen2

Description: A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017)

	Ref	Expression
Assertion	sssigagen2	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ ( sigaGen ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		simpr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ 𝐴 )
2		sssigagen	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ⊆ ( sigaGen ‘ 𝐴 ) )
3	2	adantr	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ) → 𝐴 ⊆ ( sigaGen ‘ 𝐴 ) )
4	1 3	sstrd	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ ( sigaGen ‘ 𝐴 ) )