Theorem uprcl

Description: Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025)

	Ref	Expression
Hypothesis	uprcl.c	$$⊢C={\mathrm{Base}}_E$$
Assertion	uprcl	Could not format assertion : No typesetting found for |- ( X e. ( F ( D UP E ) W ) -> ( F e. ( D Func E ) /\ W e. C ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		uprcl.c	$$⊢C={\mathrm{Base}}_E$$
2		eqid	$$⊢{\mathrm{Base}}_D={\mathrm{Base}}_D$$
3		eqid	$$⊢\mathrm{Hom}\left(D\right)=\mathrm{Hom}\left(D\right)$$
4		eqid	$$⊢\mathrm{Hom}\left(E\right)=\mathrm{Hom}\left(E\right)$$
5		eqid	$$⊢\mathrm{comp}\left(E\right)=\mathrm{comp}\left(E\right)$$
6	2 1 3 4 5	upfval	Could not format ( D UP E ) = ( f e. ( D Func E ) , w e. C |-> { <. x , m >. | ( ( x e. ( Base ` D ) /\ m e. ( w ( Hom ` E ) ( ( 1st ` f ) ` x ) ) ) /\ A. y e. ( Base ` D ) A. g e. ( w ( Hom ` E ) ( ( 1st ` f ) ` y ) ) E! k e. ( x ( Hom ` D ) y ) g = ( ( ( x ( 2nd ` f ) y ) ` k ) ( <. w , ( ( 1st ` f ) ` x ) >. ( comp ` E ) ( ( 1st ` f ) ` y ) ) m ) ) } ) : No typesetting found for |- ( D UP E ) = ( f e. ( D Func E ) , w e. C |-> { <. x , m >. | ( ( x e. ( Base ` D ) /\ m e. ( w ( Hom ` E ) ( ( 1st ` f ) ` x ) ) ) /\ A. y e. ( Base ` D ) A. g e. ( w ( Hom ` E ) ( ( 1st ` f ) ` y ) ) E! k e. ( x ( Hom ` D ) y ) g = ( ( ( x ( 2nd ` f ) y ) ` k ) ( <. w , ( ( 1st ` f ) ` x ) >. ( comp ` E ) ( ( 1st ` f ) ` y ) ) m ) ) } ) with typecode |-
7	6	elmpocl	Could not format ( X e. ( F ( D UP E ) W ) -> ( F e. ( D Func E ) /\ W e. C ) ) : No typesetting found for |- ( X e. ( F ( D UP E ) W ) -> ( F e. ( D Func E ) /\ W e. C ) ) with typecode |-