Theorem restrreld

Description: The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019)

	Ref	Expression
Hypotheses	restrreld.r	$$⊢\phi \to R\circ R\subseteq R$$
	restrreld.s	$$⊢\phi \to S={R\upharpoonright }_A$$
Assertion	restrreld	$$⊢\phi \to S\circ S\subseteq S$$

Proof

Step	Hyp	Ref	Expression
1		restrreld.r	$$⊢\phi \to R\circ R\subseteq R$$
2		restrreld.s	$$⊢\phi \to S={R\upharpoonright }_A$$
3		df-res	$$⊢{R\upharpoonright }_A=R\cap \left(A\times \mathrm{V}\right)$$
4	2 3	eqtrdi	$$⊢\phi \to S=R\cap \left(A\times \mathrm{V}\right)$$
5	1 4	xpintrreld	$$⊢\phi \to S\circ S\subseteq S$$