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Metamath Proof Explorer

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 ) = (/) )