Theorem trclsslem

Description: The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020)

	Ref	Expression
Assertion	trclsslem	$$⊢R\subseteq S\to \bigcap \left\{r|\left(R\subseteq r\wedge r\circ r\subseteq r\right)\right\}\subseteq \bigcap \left\{r|\left(S\subseteq r\wedge r\circ r\subseteq r\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		clsslem	$$⊢R\subseteq S\to \bigcap \left\{r|\left(R\subseteq r\wedge r\circ r\subseteq r\right)\right\}\subseteq \bigcap \left\{r|\left(S\subseteq r\wedge r\circ r\subseteq r\right)\right\}$$