Theorem inxpssres

Description: Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019)

	Ref	Expression
Assertion	inxpssres	$$⊢R\cap \left(A\times B\right)\subseteq {R\upharpoonright }_A$$

Proof

Step	Hyp	Ref	Expression
1		ssid	$$⊢A\subseteq A$$
2		ssv	$$⊢B\subseteq \mathrm{V}$$
3		xpss12	$$⊢\left(A\subseteq A\wedge B\subseteq \mathrm{V}\right)\to A\times B\subseteq A\times \mathrm{V}$$
4	1 2 3	mp2an	$$⊢A\times B\subseteq A\times \mathrm{V}$$
5		sslin	$$⊢A\times B\subseteq A\times \mathrm{V}\to R\cap \left(A\times B\right)\subseteq R\cap \left(A\times \mathrm{V}\right)$$
6	4 5	ax-mp	$$⊢R\cap \left(A\times B\right)\subseteq R\cap \left(A\times \mathrm{V}\right)$$
7		df-res	$$⊢{R\upharpoonright }_A=R\cap \left(A\times \mathrm{V}\right)$$
8	6 7	sseqtrri	$$⊢R\cap \left(A\times B\right)\subseteq {R\upharpoonright }_A$$