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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 ∈ A ⟼ B
Assertion fvmptn ⊢ ¬ C ∈ V → F ⁡ D = ∅

Proof

Step Hyp Ref Expression
1 fvmptn.1 ⊢ x = D → B = C
2 fvmptn.2 ⊢ F = x ∈ A ⟼ B
3 nfcv ⊢ Ⅎ _ x D
4 nfcv ⊢ Ⅎ _ x C
5 3 4 1 2 fvmptnf ⊢ ¬ C ∈ V → F ⁡ D = ∅