Theorem afv20defat

Description: If the alternate function value at an argument is the empty set, the function is defined at this argument. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv20defat	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \to F\mathrm{defAt}A$$

Proof

Step	Hyp	Ref	Expression
1		ndfatafv2	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F$$
2		pwne0	$$⊢𝒫\bigcup \mathrm{ran}F\ne \varnothing$$
3	2	neii	$$⊢\neg 𝒫\bigcup \mathrm{ran}F=\varnothing$$
4		eqeq1	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \leftrightarrow 𝒫\bigcup \mathrm{ran}F=\varnothing \right)$$
5	3 4	mtbiri	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing$$
6	1 5	syl	$$⊢\neg F\mathrm{defAt}A\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing$$
7	6	con4i	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\varnothing \to F\mathrm{defAt}A$$