Theorem up1st2ndb

Description: Combine/separate parts in the universal property predicate. (Contributed by Zhi Wang, 23-Oct-2025)

	Ref	Expression
Hypothesis	up1st2ndr.1	|- ( ph -> F e. ( D Func E ) )
Assertion	up1st2ndb	|- ( ph -> ( X ( F ( D UP E ) W ) M <-> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) )

Proof

Step	Hyp	Ref	Expression
1		up1st2ndr.1	|- ( ph -> F e. ( D Func E ) )
2		simpr	|- ( ( ph /\ X ( F ( D UP E ) W ) M ) -> X ( F ( D UP E ) W ) M )
3	2	up1st2nd	|- ( ( ph /\ X ( F ( D UP E ) W ) M ) -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M )
4	1	adantr	|- ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> F e. ( D Func E ) )
5		simpr	|- ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M )
6	4 5	up1st2ndr	|- ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> X ( F ( D UP E ) W ) M )
7	3 6	impbida	|- ( ph -> ( X ( F ( D UP E ) W ) M <-> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) )