Theorem uprcl4

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

	Ref	Expression
Hypotheses	uprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
	uprcl4.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
Assertion	uprcl4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		uprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
2		uprcl4.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
3		eqid	⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 )
4		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
5		eqid	⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 )
6		eqid	⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 )
7	1 3	uprcl3	⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐸 ) )
8	1	uprcl2	⊢ ( 𝜑 → 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 )
9	2 3 4 5 6 7 8	isuplem	⊢ ( 𝜑 → ( 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑀 ∈ ( 𝑊 ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑊 ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑋 ( Hom ‘ 𝐷 ) 𝑦 ) 𝑔 = ( ( ( 𝑋 𝐺 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑊 , ( 𝐹 ‘ 𝑋 ) ⟩ ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) 𝑀 ) ) ) )
10	1 9	mpbid	⊢ ( 𝜑 → ( ( 𝑋 ∈ 𝐵 ∧ 𝑀 ∈ ( 𝑊 ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑋 ) ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑊 ( Hom ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑋 ( Hom ‘ 𝐷 ) 𝑦 ) 𝑔 = ( ( ( 𝑋 𝐺 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑊 , ( 𝐹 ‘ 𝑋 ) ⟩ ( comp ‘ 𝐸 ) ( 𝐹 ‘ 𝑦 ) ) 𝑀 ) ) )
11	10	simplld	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )