Theorem up1st2ndb

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

	Ref	Expression
Hypothesis	up1st2ndr.1	$$⊢\phi \to F\in \left(D\mathrm{Func}E\right)$$
Assertion	up1st2ndb	Could not format assertion : No typesetting found for |- ( ph -> ( X ( F ( D UP E ) W ) M <-> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		up1st2ndr.1	$$⊢\phi \to F\in \left(D\mathrm{Func}E\right)$$
2		simpr	Could not format ( ( ph /\ X ( F ( D UP E ) W ) M ) -> X ( F ( D UP E ) W ) M ) : No typesetting found for |- ( ( ph /\ X ( F ( D UP E ) W ) M ) -> X ( F ( D UP E ) W ) M ) with typecode |-
3	2	up1st2nd	Could not format ( ( ph /\ X ( F ( D UP E ) W ) M ) -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) : No typesetting found for |- ( ( ph /\ X ( F ( D UP E ) W ) M ) -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) with typecode |-
4	1	adantr	Could not format ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> F e. ( D Func E ) ) : No typesetting found for |- ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> F e. ( D Func E ) ) with typecode |-
5		simpr	Could not format ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) : No typesetting found for |- ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) with typecode |-
6	4 5	up1st2ndr	Could not format ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> X ( F ( D UP E ) W ) M ) : No typesetting found for |- ( ( ph /\ X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) -> X ( F ( D UP E ) W ) M ) with typecode |-
7	3 6	impbida	Could not format ( ph -> ( X ( F ( D UP E ) W ) M <-> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) ) : No typesetting found for |- ( ph -> ( X ( F ( D UP E ) W ) M <-> X ( <. ( 1st ` F ) , ( 2nd ` F ) >. ( D UP E ) W ) M ) ) with typecode |-