Theorem uprcl

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

	Ref	Expression
Hypothesis	uprcl.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
Assertion	uprcl	⊢ ( 𝑋 ∈ ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) → ( 𝐹 ∈ ( 𝐷 Func 𝐸 ) ∧ 𝑊 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		uprcl.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
2		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
3		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
4		eqid	⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 )
5		eqid	⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 )
6	2 1 3 4 5	upfval	⊢ ( 𝐷 UP 𝐸 ) = ( 𝑓 ∈ ( 𝐷 Func 𝐸 ) , 𝑤 ∈ 𝐶 ↦ { ⟨ 𝑥 , 𝑚 ⟩ ∣ ( ( 𝑥 ∈ ( Base ‘ 𝐷 ) ∧ 𝑚 ∈ ( 𝑤 ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ) ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑔 ∈ ( 𝑤 ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝑓 ) ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) 𝑔 = ( ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑤 , ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝑓 ) ‘ 𝑦 ) ) 𝑚 ) ) } )
7	6	elmpocl	⊢ ( 𝑋 ∈ ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) → ( 𝐹 ∈ ( 𝐷 Func 𝐸 ) ∧ 𝑊 ∈ 𝐶 ) )