Theorem mptv

Description: Function with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013)

	Ref	Expression
Assertion	mptv	$$⊢\left(x\in \mathrm{V}⟼B\right)=\left\{⟨x,y⟩|y=B\right\}$$

Proof

Step	Hyp	Ref	Expression
1		df-mpt	$$⊢\left(x\in \mathrm{V}⟼B\right)=\left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y=B\right)\right\}$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	biantrur	$$⊢y=B\leftrightarrow \left(x\in \mathrm{V}\wedge y=B\right)$$
4	3	opabbii	$$⊢\left\{⟨x,y⟩|y=B\right\}=\left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y=B\right)\right\}$$
5	1 4	eqtr4i	$$⊢\left(x\in \mathrm{V}⟼B\right)=\left\{⟨x,y⟩|y=B\right\}$$