Theorem cic1st2ndbr

Description: Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025)

	Ref	Expression
Assertion	cic1st2ndbr	|- ( P e. ( ~=c ` C ) -> ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) )

Proof

Step	Hyp	Ref	Expression
1		cic1st2nd	|- ( P e. ( ~=c ` C ) -> P = <. ( 1st ` P ) , ( 2nd ` P ) >. )
2		id	|- ( P e. ( ~=c ` C ) -> P e. ( ~=c ` C ) )
3	1 2	eqeltrrd	|- ( P e. ( ~=c ` C ) -> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( ~=c ` C ) )
4		df-br	|- ( ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) <-> <. ( 1st ` P ) , ( 2nd ` P ) >. e. ( ~=c ` C ) )
5	3 4	sylibr	|- ( P e. ( ~=c ` C ) -> ( 1st ` P ) ( ~=c ` C ) ( 2nd ` P ) )