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Metamath Proof Explorer

Theorem pfxnndmnd

Description: The value of a prefix operation for out-of-domain arguments. (This is due to our definition of function values for out-of-domain arguments, see ndmfv ). (Contributed by AV, 3-Dec-2022) (New usage is discouraged.)

		Ref	Expression
	Assertion	pfxnndmnd	⊢ ( ¬ ( 𝑆 ∈ V ∧ 𝐿 ∈ ℕ₀ ) → ( 𝑆 prefix 𝐿 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-pfx	⊢ prefix = ( 𝑠 ∈ V , 𝑙 ∈ ℕ₀ ↦ ( 𝑠 substr ⟨ 0 , 𝑙 ⟩ ) )
2	1	mpondm0	⊢ ( ¬ ( 𝑆 ∈ V ∧ 𝐿 ∈ ℕ₀ ) → ( 𝑆 prefix 𝐿 ) = ∅ )