Theorem op1st

Description: Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004)

	Ref	Expression
Hypotheses	op1st.1	⊢ 𝐴 ∈ V
	op1st.2	⊢ 𝐵 ∈ V
Assertion	op1st	⊢ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		op1st.1	⊢ 𝐴 ∈ V
2		op1st.2	⊢ 𝐵 ∈ V
3		1stval	⊢ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = ∪ dom { ⟨ 𝐴 , 𝐵 ⟩ }
4	1 2	op1sta	⊢ ∪ dom { ⟨ 𝐴 , 𝐵 ⟩ } = 𝐴
5	3 4	eqtri	⊢ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐴