Theorem mpt0

Description: A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014)

	Ref	Expression
Assertion	mpt0	⊢ ( 𝑥 ∈ ∅ ↦ 𝐴 ) = ∅

Proof

Step	Hyp	Ref	Expression
1		ral0	⊢ ∀ 𝑥 ∈ ∅ 𝐴 ∈ V
2		eqid	⊢ ( 𝑥 ∈ ∅ ↦ 𝐴 ) = ( 𝑥 ∈ ∅ ↦ 𝐴 )
3	2	fnmpt	⊢ ( ∀ 𝑥 ∈ ∅ 𝐴 ∈ V → ( 𝑥 ∈ ∅ ↦ 𝐴 ) Fn ∅ )
4	1 3	ax-mp	⊢ ( 𝑥 ∈ ∅ ↦ 𝐴 ) Fn ∅
5		fn0	⊢ ( ( 𝑥 ∈ ∅ ↦ 𝐴 ) Fn ∅ ↔ ( 𝑥 ∈ ∅ ↦ 𝐴 ) = ∅ )
6	4 5	mpbi	⊢ ( 𝑥 ∈ ∅ ↦ 𝐴 ) = ∅