Theorem dfatafv2eqfv

Description: If a function is defined at a class A , the alternate function value equals the function's value at A . (Contributed by AV, 3-Sep-2022)

	Ref	Expression
Assertion	dfatafv2eqfv	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfafv22	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=if\left(F\mathrm{defAt}A,F\left(A\right),𝒫\bigcup \mathrm{ran}F\right)$$
2		iftrue	$$⊢F\mathrm{defAt}A\to if\left(F\mathrm{defAt}A,F\left(A\right),𝒫\bigcup \mathrm{ran}F\right)=F\left(A\right)$$
3	1 2	eqtrid	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$