Theorem carsgcl

Description: Closure of the Caratheodory measurable sets. (Contributed by Thierry Arnoux, 17-May-2020)

	Ref	Expression
Hypotheses	carsgval.1	|- ( ph -> O e. V )
	carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
Assertion	carsgcl	|- ( ph -> ( toCaraSiga ` M ) C_ ~P O )

Proof

Step	Hyp	Ref	Expression
1		carsgval.1	|- ( ph -> O e. V )
2		carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
3	1 2	carsgval	|- ( ph -> ( toCaraSiga ` M ) = { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } )
4		ssrab2	|- { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } C_ ~P O
5	3 4	eqsstrdi	|- ( ph -> ( toCaraSiga ` M ) C_ ~P O )