Theorem fundmafv2rnb

Description: The alternate function value at a class A is defined, i.e., in the range of the function iff A is in the domain of the function. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	fundmafv2rnb	$$⊢\mathrm{Fun}F\to \left(A\in \mathrm{dom}F\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\right)$$

Proof

Step	Hyp	Ref	Expression
1		funres	$$⊢\mathrm{Fun}F\to \mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)$$
2		dmafv2rnb	$$⊢\mathrm{Fun}\left({F\upharpoonright }_{\left\{A\right\}}\right)\to \left(A\in \mathrm{dom}F\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\right)$$
3	1 2	syl	$$⊢\mathrm{Fun}F\to \left(A\in \mathrm{dom}F\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\right)$$