Theorem afvnufveq

Description: The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvnufveq	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\ne \mathrm{V}\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		afvfundmfveq	$$⊢F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$
2		afvnfundmuv	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$
3	1 2	nsyl5	$$⊢\neg \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$
4	3	necon1ai	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\ne \mathrm{V}\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$