Theorem uprcl5

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

	Ref	Expression
Hypotheses	uprcl2.x	No typesetting found for |- ( ph -> X ( <. F , G >. ( D UP E ) W ) M ) with typecode |-
	uprcl5.j	$$⊢J=\mathrm{Hom}\left(E\right)$$
Assertion	uprcl5	$$⊢\phi \to M\in \left(WJF\left(X\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		uprcl2.x	Could not format ( ph -> X ( <. F , G >. ( D UP E ) W ) M ) : No typesetting found for |- ( ph -> X ( <. F , G >. ( D UP E ) W ) M ) with typecode |-
2		uprcl5.j	$$⊢J=\mathrm{Hom}\left(E\right)$$
3		eqid	$$⊢{\mathrm{Base}}_D={\mathrm{Base}}_D$$
4		eqid	$$⊢{\mathrm{Base}}_E={\mathrm{Base}}_E$$
5		eqid	$$⊢\mathrm{Hom}\left(D\right)=\mathrm{Hom}\left(D\right)$$
6		eqid	$$⊢\mathrm{comp}\left(E\right)=\mathrm{comp}\left(E\right)$$
7	1 4	uprcl3	$$⊢\phi \to W\in {\mathrm{Base}}_E$$
8	1	uprcl2	$$⊢\phi \to F\left(D\mathrm{Func}E\right)G$$
9	3 4 5 2 6 7 8	isuplem	Could not format ( ph -> ( X ( <. F , G >. ( D UP E ) W ) M <-> ( ( X e. ( Base ` D ) /\ M e. ( W J ( F ` X ) ) ) /\ A. y e. ( Base ` D ) A. g e. ( W J ( F ` y ) ) E! k e. ( X ( Hom ` D ) y ) g = ( ( ( X G y ) ` k ) ( <. W , ( F ` X ) >. ( comp ` E ) ( F ` y ) ) M ) ) ) ) : No typesetting found for |- ( ph -> ( X ( <. F , G >. ( D UP E ) W ) M <-> ( ( X e. ( Base ` D ) /\ M e. ( W J ( F ` X ) ) ) /\ A. y e. ( Base ` D ) A. g e. ( W J ( F ` y ) ) E! k e. ( X ( Hom ` D ) y ) g = ( ( ( X G y ) ` k ) ( <. W , ( F ` X ) >. ( comp ` E ) ( F ` y ) ) M ) ) ) ) with typecode |-
10	1 9	mpbid	$$⊢\phi \to \left(\left(X\in {\mathrm{Base}}_D\wedge M\in \left(WJF\left(X\right)\right)\right)\wedge \forall y\in {\mathrm{Base}}_D\forall g\in \left(WJF\left(y\right)\right)\exists !k\in \left(X\mathrm{Hom}\left(D\right)y\right)g=\left(XGy\right)\left(k\right)\left(⟨W,F\left(X\right)⟩\mathrm{comp}\left(E\right)F\left(y\right)\right)M\right)$$
11	10	simplrd	$$⊢\phi \to M\in \left(WJF\left(X\right)\right)$$