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Metamath Proof Explorer

Theorem ndmima

Description: The image of a singleton outside the domain is empty. (Contributed by NM, 22-May-1998) (Proof shortened by OpenAI, 3-Jul-2020)

		Ref	Expression
	Assertion	ndmima	⊢ ( ¬ 𝐴 ∈ dom 𝐵 → ( 𝐵 “ { 𝐴 } ) = ∅ )

Step	Hyp	Ref	Expression
1		imadisj	⊢ ( ( 𝐵 “ { 𝐴 } ) = ∅ ↔ ( dom 𝐵 ∩ { 𝐴 } ) = ∅ )
2		disjsn	⊢ ( ( dom 𝐵 ∩ { 𝐴 } ) = ∅ ↔ ¬ 𝐴 ∈ dom 𝐵 )
3	1 2	sylbbr	⊢ ( ¬ 𝐴 ∈ dom 𝐵 → ( 𝐵 “ { 𝐴 } ) = ∅ )