Theorem afvvfveq

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	afvvfveq	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		nvelim	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}\to \neg \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B$$
2	1	necon2ai	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\ne \mathrm{V}$$
3		afvnufveq	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\ne \mathrm{V}\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$
4	2 3	syl	$$⊢\left(F\mathrm{\text{'}\text{'}\text{'}}A\right)\in B\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=F\left(A\right)$$