Theorem xpssres

Description: Restriction of a constant function (or other Cartesian product). (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		df-res	$$⊢{\left(A\times B\right)\upharpoonright }_C=\left(A\times B\right)\cap \left(C\times \mathrm{V}\right)$$
2		inxp	$$⊢\left(A\times B\right)\cap \left(C\times \mathrm{V}\right)=\left(A\cap C\right)\times \left(B\cap \mathrm{V}\right)$$
3		inv1	$$⊢B\cap \mathrm{V}=B$$
4	3	xpeq2i	$$⊢\left(A\cap C\right)\times \left(B\cap \mathrm{V}\right)=\left(A\cap C\right)\times B$$
5	1 2 4	3eqtri	$$⊢{\left(A\times B\right)\upharpoonright }_C=\left(A\cap C\right)\times B$$
6		sseqin2	$$⊢C\subseteq A\leftrightarrow A\cap C=C$$
7	6	biimpi	$$⊢C\subseteq A\to A\cap C=C$$
8	7	xpeq1d	$$⊢C\subseteq A\to \left(A\cap C\right)\times B=C\times B$$
9	5 8	syl5eq	$$⊢C\subseteq A\to {\left(A\times B\right)\upharpoonright }_C=C\times B$$