Theorem sgsiga

Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017)

	Ref	Expression
Hypothesis	sgsiga.1	|- ( ph -> A e. V )
Assertion	sgsiga	|- ( ph -> ( sigaGen ` A ) e. U. ran sigAlgebra )

Proof

Step	Hyp	Ref	Expression
1		sgsiga.1	|- ( ph -> A e. V )
2		sigagensiga	|- ( A e. V -> ( sigaGen ` A ) e. ( sigAlgebra ` U. A ) )
3		elrnsiga	|- ( ( sigaGen ` A ) e. ( sigAlgebra ` U. A ) -> ( sigaGen ` A ) e. U. ran sigAlgebra )
4	1 2 3	3syl	|- ( ph -> ( sigaGen ` A ) e. U. ran sigAlgebra )