Theorem mpt0

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

	Ref	Expression
Assertion	mpt0	$$⊢\left(x\in \varnothing ⟼A\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		ral0	$$⊢\forall x\in \varnothing A\in \mathrm{V}$$
2		eqid	$$⊢\left(x\in \varnothing ⟼A\right)=\left(x\in \varnothing ⟼A\right)$$
3	2	fnmpt	$$⊢\forall x\in \varnothing A\in \mathrm{V}\to \left(x\in \varnothing ⟼A\right)Fn\varnothing$$
4	1 3	ax-mp	$$⊢\left(x\in \varnothing ⟼A\right)Fn\varnothing$$
5		fn0	$$⊢\left(x\in \varnothing ⟼A\right)Fn\varnothing \leftrightarrow \left(x\in \varnothing ⟼A\right)=\varnothing$$
6	4 5	mpbi	$$⊢\left(x\in \varnothing ⟼A\right)=\varnothing$$