Theorem relrn0

Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004)

	Ref	Expression
Assertion	relrn0	$$⊢\mathrm{Rel}A\to \left(A=\varnothing \leftrightarrow \mathrm{ran}A=\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		reldm0	$$⊢\mathrm{Rel}A\to \left(A=\varnothing \leftrightarrow \mathrm{dom}A=\varnothing \right)$$
2		dm0rn0	$$⊢\mathrm{dom}A=\varnothing \leftrightarrow \mathrm{ran}A=\varnothing$$
3	1 2	syl6bb	$$⊢\mathrm{Rel}A\to \left(A=\varnothing \leftrightarrow \mathrm{ran}A=\varnothing \right)$$