Theorem tz6.12-2-afv2

Description: Function value when F is (locally) not a function. Theorem 6.12(2) of TakeutiZaring p. 27, analogous to tz6.12-2 . (Contributed by AV, 5-Sep-2022)

	Ref	Expression
Assertion	tz6.12-2-afv2	$$⊢\neg \exists !xAFx\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		dfdfat2	$$⊢F\mathrm{defAt}A\leftrightarrow \left(A\in \mathrm{dom}F\wedge \exists !xAFx\right)$$
2	1	simprbi	$$⊢F\mathrm{defAt}A\to \exists !xAFx$$
3	2	con3i	$$⊢\neg \exists !xAFx\to \neg F\mathrm{defAt}A$$
4		ndfatafv2nrn	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$
5	3 4	syl	$$⊢\neg \exists !xAFx\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$