Theorem fvmptn

Description: This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class C it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg . (Contributed by NM, 21-Oct-2003) (Revised by Mario Carneiro, 9-Sep-2013)

	Ref	Expression
Hypotheses	fvmptn.1	$$⊢x=D\to B=C$$
	fvmptn.2	$$⊢F=\left(x\in A⟼B\right)$$
Assertion	fvmptn	$$⊢\neg C\in \mathrm{V}\to F\left(D\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		fvmptn.1	$$⊢x=D\to B=C$$
2		fvmptn.2	$$⊢F=\left(x\in A⟼B\right)$$
3		nfcv	$$⊢\underset{\_}{Ⅎ}xD$$
4		nfcv	$$⊢\underset{\_}{Ⅎ}xC$$
5	3 4 1 2	fvmptnf	$$⊢\neg C\in \mathrm{V}\to F\left(D\right)=\varnothing$$