Theorem clsslem

Description: The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020)

	Ref	Expression
Assertion	clsslem	|- ( R C_ S -> |^| { r | ( R C_ r /\ ph ) } C_ |^| { r | ( S C_ r /\ ph ) } )

Proof

Step	Hyp	Ref	Expression
1		sstr2	|- ( R C_ S -> ( S C_ r -> R C_ r ) )
2	1	anim1d	|- ( R C_ S -> ( ( S C_ r /\ ph ) -> ( R C_ r /\ ph ) ) )
3	2	ss2abdv	|- ( R C_ S -> { r | ( S C_ r /\ ph ) } C_ { r | ( R C_ r /\ ph ) } )
4		intss	|- ( { r | ( S C_ r /\ ph ) } C_ { r | ( R C_ r /\ ph ) } -> |^| { r | ( R C_ r /\ ph ) } C_ |^| { r | ( S C_ r /\ ph ) } )
5	3 4	syl	|- ( R C_ S -> |^| { r | ( R C_ r /\ ph ) } C_ |^| { r | ( S C_ r /\ ph ) } )