Theorem ot21std

Description: Extract the first member of an ordered triple. Deduction version. (Contributed by Scott Fenton, 21-Aug-2024)

	Ref	Expression
Hypotheses	ot21st.1	⊢ 𝐴 ∈ V
	ot21st.2	⊢ 𝐵 ∈ V
	ot21st.3	⊢ 𝐶 ∈ V
Assertion	ot21std	⊢ ( 𝑋 = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ → ( 1st ‘ ( 1st ‘ 𝑋 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ot21st.1	⊢ 𝐴 ∈ V
2		ot21st.2	⊢ 𝐵 ∈ V
3		ot21st.3	⊢ 𝐶 ∈ V
4		opex	⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ V
5	4 3	op1std	⊢ ( 𝑋 = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ → ( 1st ‘ 𝑋 ) = ⟨ 𝐴 , 𝐵 ⟩ )
6	5	fveq2d	⊢ ( 𝑋 = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ → ( 1st ‘ ( 1st ‘ 𝑋 ) ) = ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) )
7	1 2	op1st	⊢ ( 1st ‘ ⟨ 𝐴 , 𝐵 ⟩ ) = 𝐴
8	6 7	eqtrdi	⊢ ( 𝑋 = ⟨ ⟨ 𝐴 , 𝐵 ⟩ , 𝐶 ⟩ → ( 1st ‘ ( 1st ‘ 𝑋 ) ) = 𝐴 )