Theorem afv2rnfveq

Description: If the alternate function value is defined, i.e., in the range of the function, the alternate function value equals the function's value. (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	afv2rnfveq	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfatafv2rnb	$$⊢F\mathrm{defAt}A\leftrightarrow \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F$$
2		dfatafv2eqfv	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$
3	1 2	sylbir	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)\in \mathrm{ran}F\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$