Theorem afv2elrn

Description: An alternate function value belongs to the range of the function, analogous to fvelrn . (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv2elrn	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		fundmdfat	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to F\mathrm{defAt}A$$
2		dfatafv2rnb	$$⊢F\mathrm{defAt}A\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
3	1 2	sylib	$$⊢\left(\mathrm{Fun}F\wedge A\in \mathrm{dom}F\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$