Theorem ndfatafv2undef

Description: The alternate function value at a class A is undefined if the function, whose range is a set, is not defined at A . (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	ndfatafv2undef	$$⊢\left(\mathrm{ran}F\in V\wedge \neg F\mathrm{defAt}A\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\mathrm{Undef}\left(\mathrm{ran}F\right)$$

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		undefval	$$⊢\mathrm{ran}F\in V\to \mathrm{Undef}\left(\mathrm{ran}F\right)=𝒫\bigcup \mathrm{ran}F$$
3	2	eqcomd	$$⊢\mathrm{ran}F\in V\to 𝒫\bigcup \mathrm{ran}F=\mathrm{Undef}\left(\mathrm{ran}F\right)$$
4	1 3	sylan9eqr	$$⊢\left(\mathrm{ran}F\in V\wedge \neg F\mathrm{defAt}A\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=\mathrm{Undef}\left(\mathrm{ran}F\right)$$