Theorem fssxp

Description: A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	fssxp	$$⊢F:A⟶B\to F\subseteq A\times B$$

Proof

Step	Hyp	Ref	Expression
1		frel	$$⊢F:A⟶B\to \mathrm{Rel}F$$
2		relssdmrn	$$⊢\mathrm{Rel}F\to F\subseteq \mathrm{dom}F\times \mathrm{ran}F$$
3	1 2	syl	$$⊢F:A⟶B\to F\subseteq \mathrm{dom}F\times \mathrm{ran}F$$
4		fdm	$$⊢F:A⟶B\to \mathrm{dom}F=A$$
5		eqimss	$$⊢\mathrm{dom}F=A\to \mathrm{dom}F\subseteq A$$
6	4 5	syl	$$⊢F:A⟶B\to \mathrm{dom}F\subseteq A$$
7		frn	$$⊢F:A⟶B\to \mathrm{ran}F\subseteq B$$
8		xpss12	$$⊢\left(\mathrm{dom}F\subseteq A\wedge \mathrm{ran}F\subseteq B\right)\to \mathrm{dom}F\times \mathrm{ran}F\subseteq A\times B$$
9	6 7 8	syl2anc	$$⊢F:A⟶B\to \mathrm{dom}F\times \mathrm{ran}F\subseteq A\times B$$
10	3 9	sstrd	$$⊢F:A⟶B\to F\subseteq A\times B$$