Theorem dmsnop

Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Hypothesis	dmsnop.1	$$⊢B\in \mathrm{V}$$
Assertion	dmsnop	$$⊢\mathrm{dom}\left\{⟨A,B⟩\right\}=\left\{A\right\}$$

Proof

Step	Hyp	Ref	Expression
1		dmsnop.1	$$⊢B\in \mathrm{V}$$
2		dmsnopg	$$⊢B\in \mathrm{V}\to \mathrm{dom}\left\{⟨A,B⟩\right\}=\left\{A\right\}$$
3	1 2	ax-mp	$$⊢\mathrm{dom}\left\{⟨A,B⟩\right\}=\left\{A\right\}$$