Theorem nfunsnafv2

Description: If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn . (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	nfunsnafv2	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		olc	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(\neg A\in \mathrm{dom}F\vee \neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)$$
2		ianor	$$⊢\neg \left(A\in \mathrm{dom}F\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)\leftrightarrow \left(\neg A\in \mathrm{dom}F\vee \neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)$$
3		df-dfat	$$⊢F\mathrm{defAt}A\leftrightarrow \left(A\in \mathrm{dom}F\wedge \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)$$
4	2 3	xchnxbir	$$⊢\neg F\mathrm{defAt}A\leftrightarrow \left(\neg A\in \mathrm{dom}F\vee \neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\right)$$
5	1 4	sylibr	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \neg F\mathrm{defAt}A$$
6		ndfatafv2nrn	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$
7	5 6	syl	$$⊢\neg \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$