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 ⟩ → ( 1st ‘ 𝐶 ) = 𝐴 )

Proof

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