Theorem ima0

Description: Image of the empty set. Theorem 3.16(ii) of Monk1 p. 38. (Contributed by NM, 20-May-1998)

	Ref	Expression
Assertion	ima0	$$⊢A\left[\varnothing \right]=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		df-ima	$$⊢A\left[\varnothing \right]=\mathrm{ran}\left({A\upharpoonright }_{\varnothing }\right)$$
2		res0	$$⊢{A\upharpoonright }_{\varnothing }=\varnothing$$
3	2	rneqi	$$⊢\mathrm{ran}\left({A\upharpoonright }_{\varnothing }\right)=\mathrm{ran}\varnothing$$
4		rn0	$$⊢\mathrm{ran}\varnothing =\varnothing$$
5	1 3 4	3eqtri	$$⊢A\left[\varnothing \right]=\varnothing$$