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	$$⊢\phi \to A\subseteq B$$
	coss12d.c	$$⊢\phi \to C\subseteq D$$
Assertion	coss12d	$$⊢\phi \to A\circ C\subseteq B\circ D$$

Proof

Step	Hyp	Ref	Expression
1		coss12d.a	$$⊢\phi \to A\subseteq B$$
2		coss12d.c	$$⊢\phi \to C\subseteq D$$
3		coss1	$$⊢A\subseteq B\to A\circ C\subseteq B\circ C$$
4	1 3	syl	$$⊢\phi \to A\circ C\subseteq B\circ C$$
5		coss2	$$⊢C\subseteq D\to B\circ C\subseteq B\circ D$$
6	2 5	syl	$$⊢\phi \to B\circ C\subseteq B\circ D$$
7	4 6	sstrd	$$⊢\phi \to A\circ C\subseteq B\circ D$$