Theorem idresssidinxp

Description: Condition for the identity restriction to be a subclass of identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018)

	Ref	Expression
Assertion	idresssidinxp	$$⊢A\subseteq B\to {\mathrm{I}\upharpoonright }_A\subseteq \mathrm{I}\cap \left(A\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		resss	$$⊢{\mathrm{I}\upharpoonright }_A\subseteq \mathrm{I}$$
2	1	a1i	$$⊢A\subseteq B\to {\mathrm{I}\upharpoonright }_A\subseteq \mathrm{I}$$
3		idssxp	$$⊢{\mathrm{I}\upharpoonright }_A\subseteq A\times A$$
4		xpss2	$$⊢A\subseteq B\to A\times A\subseteq A\times B$$
5	3 4	sstrid	$$⊢A\subseteq B\to {\mathrm{I}\upharpoonright }_A\subseteq A\times B$$
6	2 5	ssind	$$⊢A\subseteq B\to {\mathrm{I}\upharpoonright }_A\subseteq \mathrm{I}\cap \left(A\times B\right)$$