Theorem ndfatafv2nrn

Description: The alternate function value at a class A at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	ndfatafv2nrn	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$

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		pwuninel	$$⊢\neg 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F$$
3		df-nel	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\leftrightarrow \neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
4		eleq1	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\leftrightarrow 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F\right)$$
5	4	notbid	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\leftrightarrow \neg 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F\right)$$
6	3 5	syl5bb	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\leftrightarrow \neg 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F\right)$$
7	2 6	mpbiri	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$
8	1 7	syl	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$