Theorem afvfundmfveq

Description: If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017)

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

Proof

Step	Hyp	Ref	Expression
1		dfafv2	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=if\left(F\mathrm{defAt}A,F\left(A\right),\mathrm{V}\right)$$
2		iftrue	$$⊢F\mathrm{defAt}A\to if\left(F\mathrm{defAt}A,F\left(A\right),\mathrm{V}\right)=F\left(A\right)$$
3	1 2	eqtrid	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$