Theorem ot22ndd

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

	Ref	Expression
Hypotheses	ot21st.1	|- A e. _V
	ot21st.2	|- B e. _V
	ot21st.3	|- C e. _V
Assertion	ot22ndd	|- ( X = <. <. A , B >. , C >. -> ( 2nd ` ( 1st ` X ) ) = B )

Proof

Step	Hyp	Ref	Expression
1		ot21st.1	|- A e. _V
2		ot21st.2	|- B e. _V
3		ot21st.3	|- C e. _V
4		opex	|- <. A , B >. e. _V
5	4 3	op1std	|- ( X = <. <. A , B >. , C >. -> ( 1st ` X ) = <. A , B >. )
6	5	fveq2d	|- ( X = <. <. A , B >. , C >. -> ( 2nd ` ( 1st ` X ) ) = ( 2nd ` <. A , B >. ) )
7	1 2	op2nd	|- ( 2nd ` <. A , B >. ) = B
8	6 7	eqtrdi	|- ( X = <. <. A , B >. , C >. -> ( 2nd ` ( 1st ` X ) ) = B )