Theorem dfatafv2rnb

Description: The alternate function value at a class A is defined, i.e. in the range of the function, iff the function is defined at A . (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	dfatafv2rnb	$$⊢F\mathrm{defAt}A\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		funressndmafv2rn	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
2		ndfatafv2nrn	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\notin \mathrm{ran}F$$
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	2 3	sylib	$$⊢\neg F\mathrm{defAt}A\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
5	4	con4i	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\to F\mathrm{defAt}A$$
6	1 5	impbii	$$⊢F\mathrm{defAt}A\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$