Theorem fvmptndm

Description: Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020)

	Ref	Expression
Hypothesis	fvmptndm.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	fvmptndm	⊢ ( ¬ 𝑋 ∈ 𝐴 → ( 𝐹 ‘ 𝑋 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		fvmptndm.1	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
3	1 2	eqtri	⊢ 𝐹 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) }
4	3	fvopab4ndm	⊢ ( ¬ 𝑋 ∈ 𝐴 → ( 𝐹 ‘ 𝑋 ) = ∅ )