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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 ) = ∅ )