Theorem coss12d

Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019) (Proof shortened by BJ, 8-Nov-2026)

	Ref	Expression
Hypotheses	coss12d.a	|- ( ph -> A C_ B )
	coss12d.c	|- ( ph -> C C_ D )
Assertion	coss12d	|- ( ph -> ( A o. C ) C_ ( B o. D ) )

Proof

Step	Hyp	Ref	Expression
1		coss12d.a	|- ( ph -> A C_ B )
2		coss12d.c	|- ( ph -> C C_ D )
3		coss1	|- ( A C_ B -> ( A o. C ) C_ ( B o. C ) )
4	1 3	syl	|- ( ph -> ( A o. C ) C_ ( B o. C ) )
5		coss2	|- ( C C_ D -> ( B o. C ) C_ ( B o. D ) )
6	2 5	syl	|- ( ph -> ( B o. C ) C_ ( B o. D ) )
7	4 6	sstrd	|- ( ph -> ( A o. C ) C_ ( B o. D ) )