Theorem rnmpt0f

Description: The range of a function in maps-to notation is empty if and only if its domain is empty. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	rnmpt0f.1	$$⊢Ⅎx\phi$$
	rnmpt0f.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
	rnmpt0f.3	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	rnmpt0f	$$⊢\phi \to \left(\mathrm{ran}F=\varnothing \leftrightarrow A=\varnothing \right)$$

Proof

Step	Hyp	Ref	Expression
1		rnmpt0f.1	$$⊢Ⅎx\phi$$
2		rnmpt0f.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
3		rnmpt0f.3	$$⊢F=\left(x\in A⟼B\right)$$
4	2	ex	$$⊢\phi \to \left(x\in A\to B\in V\right)$$
5	1 4	ralrimi	$$⊢\phi \to \forall x\in AB\in V$$
6		dmmptg	$$⊢\forall x\in AB\in V\to \mathrm{dom}\left(x\in A⟼B\right)=A$$
7	5 6	syl	$$⊢\phi \to \mathrm{dom}\left(x\in A⟼B\right)=A$$
8	7	eqcomd	$$⊢\phi \to A=\mathrm{dom}\left(x\in A⟼B\right)$$
9	8	eqeq1d	$$⊢\phi \to \left(A=\varnothing \leftrightarrow \mathrm{dom}\left(x\in A⟼B\right)=\varnothing \right)$$
10		dm0rn0	$$⊢\mathrm{dom}\left(x\in A⟼B\right)=\varnothing \leftrightarrow \mathrm{ran}\left(x\in A⟼B\right)=\varnothing$$
11	10	a1i	$$⊢\phi \to \left(\mathrm{dom}\left(x\in A⟼B\right)=\varnothing \leftrightarrow \mathrm{ran}\left(x\in A⟼B\right)=\varnothing \right)$$
12	3	rneqi	$$⊢\mathrm{ran}F=\mathrm{ran}\left(x\in A⟼B\right)$$
13	12	a1i	$$⊢\phi \to \mathrm{ran}F=\mathrm{ran}\left(x\in A⟼B\right)$$
14	13	eqcomd	$$⊢\phi \to \mathrm{ran}\left(x\in A⟼B\right)=\mathrm{ran}F$$
15	14	eqeq1d	$$⊢\phi \to \left(\mathrm{ran}\left(x\in A⟼B\right)=\varnothing \leftrightarrow \mathrm{ran}F=\varnothing \right)$$
16	9 11 15	3bitrrd	$$⊢\phi \to \left(\mathrm{ran}F=\varnothing \leftrightarrow A=\varnothing \right)$$