Theorem afvnfundmuv

Description: If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017)

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

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		iffalse	$$⊢\neg F\mathrm{defAt}A\to if\left(F\mathrm{defAt}A,F\left(A\right),\mathrm{V}\right)=\mathrm{V}$$
3	1 2	syl5eq	$$⊢\neg F\mathrm{defAt}A\to \left(F\mathrm{\text{'}\text{'}\text{'}}A\right)=\mathrm{V}$$