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Metamath Proof Explorer

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 )