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Metamath Proof Explorer

Theorem mpo0v

Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015) (Revised by Mario Carneiro, 15-May-2015) (Proof shortened by AV, 27-Jan-2024)

		Ref	Expression
	Assertion	mpo0v	⊢ ( 𝑥 ∈ ∅ , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∅ = ∅
2	1	orci	⊢ ( ∅ = ∅ ∨ 𝐵 = ∅ )
3		0mpo0	⊢ ( ( ∅ = ∅ ∨ 𝐵 = ∅ ) → ( 𝑥 ∈ ∅ , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ∅ )
4	2 3	ax-mp	⊢ ( 𝑥 ∈ ∅ , 𝑦 ∈ 𝐵 ↦ 𝐶 ) = ∅