Theorem afv2ndefb

Description: Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022)

	Ref	Expression
Assertion	afv2ndefb	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		pwuninel	$$⊢\neg 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F$$
2		df-nel	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\leftrightarrow \neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
3		eleq1	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\leftrightarrow 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F\right)$$
4	3	notbid	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\leftrightarrow \neg 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F\right)$$
5	2 4	syl5bb	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\leftrightarrow \neg 𝒫\bigcup \mathrm{ran}F\in \mathrm{ran}F\right)$$
6	1 5	mpbiri	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$
7		funressndmafv2rn	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
8	7	con3i	$$⊢\neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\to \neg F\mathrm{defAt}A$$
9	2 8	sylbi	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\to \neg F\mathrm{defAt}A$$
10		ndfatafv2	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F$$
11	9 10	syl	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F$$
12	6 11	impbii	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=𝒫\bigcup \mathrm{ran}F\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$