Metamath Proof Explorer


Theorem op1std

Description: Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015)

Ref Expression
Hypotheses op1st.1 𝐴 ∈ V
op1st.2 𝐵 ∈ V
Assertion op1std ( 𝐶 = ⟨ 𝐴 , 𝐵 ⟩ → ( 1st𝐶 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 op1st.1 𝐴 ∈ V
2 op1st.2 𝐵 ∈ V
3 fveq2 ( 𝐶 = ⟨ 𝐴 , 𝐵 ⟩ → ( 1st𝐶 ) = ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
4 1 2 op1st ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐴
5 3 4 eqtrdi ( 𝐶 = ⟨ 𝐴 , 𝐵 ⟩ → ( 1st𝐶 ) = 𝐴 )