Theorem coss12d

Description: Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019)

	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	2	ssbrd	$$⊢\phi \to \left(xCy\to xDy\right)$$
4	1	ssbrd	$$⊢\phi \to \left(yAz\to yBz\right)$$
5	3 4	anim12d	$$⊢\phi \to \left(\left(xCy\wedge yAz\right)\to \left(xDy\wedge yBz\right)\right)$$
6	5	eximdv	$$⊢\phi \to \left(\exists y\left(xCy\wedge yAz\right)\to \exists y\left(xDy\wedge yBz\right)\right)$$
7	6	ssopab2dv	$$⊢\phi \to \left\{⟨x,z⟩|\exists y\left(xCy\wedge yAz\right)\right\}\subseteq \left\{⟨x,z⟩|\exists y\left(xDy\wedge yBz\right)\right\}$$
8		df-co	$$⊢A\circ C=\left\{⟨x,z⟩|\exists y\left(xCy\wedge yAz\right)\right\}$$
9		df-co	$$⊢B\circ D=\left\{⟨x,z⟩|\exists y\left(xDy\wedge yBz\right)\right\}$$
10	7 8 9	3sstr4g	$$⊢\phi \to A\circ C\subseteq B\circ D$$