This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )