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 -> B = C )
	fvmptn.2	|- F = ( x e. A |-> B )
Assertion	fvmptn	|- ( -. C e. _V -> ( F ` D ) = (/) )

Proof

Step	Hyp	Ref	Expression
1		fvmptn.1	|- ( x = D -> B = C )
2		fvmptn.2	|- F = ( x e. A |-> B )
3		nfcv	|- F/_ x D
4		nfcv	|- F/_ x C
5	3 4 1 2	fvmptnf	|- ( -. C e. _V -> ( F ` D ) = (/) )