Theorem xpssres

Description: Restriction of a Cartesian product to a subclass of the first factor. (Contributed by Stefan O'Rear, 24-Jan-2015)

	Ref	Expression
Assertion	xpssres	$$⊢C\subseteq A\to {\left(A\times B\right)\upharpoonright }_C=C\times B$$

Proof

Step	Hyp	Ref	Expression
1		xpres	$$⊢{\left(A\times B\right)\upharpoonright }_C=\left(A\cap C\right)\times B$$
2		sseqin2	$$⊢C\subseteq A\leftrightarrow A\cap C=C$$
3	2	biimpi	$$⊢C\subseteq A\to A\cap C=C$$
4	3	xpeq1d	$$⊢C\subseteq A\to \left(A\cap C\right)\times B=C\times B$$
5	1 4	eqtrid	$$⊢C\subseteq A\to {\left(A\times B\right)\upharpoonright }_C=C\times B$$