Theorem xpintrreld

Description: The intersection of a transitive relation with a Cartesian product is a transitive relation. (Contributed by RP, 24-Dec-2019)

	Ref	Expression
Hypotheses	xpintrreld.r	$$⊢\phi \to R\circ R\subseteq R$$
	xpintrreld.s	$$⊢\phi \to S=R\cap \left(A\times B\right)$$
Assertion	xpintrreld	$$⊢\phi \to S\circ S\subseteq S$$

Proof

Step	Hyp	Ref	Expression
1		xpintrreld.r	$$⊢\phi \to R\circ R\subseteq R$$
2		xpintrreld.s	$$⊢\phi \to S=R\cap \left(A\times B\right)$$
3		xptrrel	$$⊢\left(A\times B\right)\circ \left(A\times B\right)\subseteq A\times B$$
4	3	a1i	$$⊢\phi \to \left(A\times B\right)\circ \left(A\times B\right)\subseteq A\times B$$
5	1 4 2	trrelind	$$⊢\phi \to S\circ S\subseteq S$$