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 ∈ V
Assertion	dmsnop	⊢ dom { ⟨ 𝐴 , 𝐵 ⟩ } = { 𝐴 }

Proof

Step	Hyp	Ref	Expression
1		dmsnop.1	⊢ 𝐵 ∈ V
2		dmsnopg	⊢ ( 𝐵 ∈ V → dom { ⟨ 𝐴 , 𝐵 ⟩ } = { 𝐴 } )
3	1 2	ax-mp	⊢ dom { ⟨ 𝐴 , 𝐵 ⟩ } = { 𝐴 }