This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem isup2

Description: The universal property of a universal pair. (Contributed by Zhi Wang, 24-Sep-2025)

Ref Expression
Hypotheses isup2.b ⊢ 𝐵 = ( Base ‘ 𝐷 )
isup2.h ⊢ 𝐻 = ( Hom ‘ 𝐷 )
isup2.j ⊢ 𝐽 = ( Hom ‘ 𝐸 )
isup2.o ⊢ 𝑂 = ( comp ‘ 𝐸 )
isup2.x ⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
Assertion isup2 ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑋 𝐻 𝑦 ) 𝑔 = ( ( ( 𝑋 𝐺 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑊 , ( 𝐹 ‘ 𝑋 ) ⟩ 𝑂 ( 𝐹 ‘ 𝑦 ) ) 𝑀 ) )

Proof

Step Hyp Ref Expression
1 isup2.b ⊢ 𝐵 = ( Base ‘ 𝐷 )
2 isup2.h ⊢ 𝐻 = ( Hom ‘ 𝐷 )
3 isup2.j ⊢ 𝐽 = ( Hom ‘ 𝐸 )
4 isup2.o ⊢ 𝑂 = ( comp ‘ 𝐸 )
5 isup2.x ⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
6 eqid ⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 )
7 5 6 uprcl3 ⊢ ( 𝜑 → 𝑊 ∈ ( Base ‘ 𝐸 ) )
8 5 uprcl2 ⊢ ( 𝜑 → 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 )
9 5 1 uprcl4 ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
10 5 3 uprcl5 ⊢ ( 𝜑 → 𝑀 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑋 ) ) )
11 1 6 2 3 4 7 8 9 10 isup ⊢ ( 𝜑 → ( 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 ↔ ∀ 𝑦 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑋 𝐻 𝑦 ) 𝑔 = ( ( ( 𝑋 𝐺 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑊 , ( 𝐹 ‘ 𝑋 ) ⟩ 𝑂 ( 𝐹 ‘ 𝑦 ) ) 𝑀 ) ) )
12 5 11 mpbid ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑋 𝐻 𝑦 ) 𝑔 = ( ( ( 𝑋 𝐺 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑊 , ( 𝐹 ‘ 𝑋 ) ⟩ 𝑂 ( 𝐹 ‘ 𝑦 ) ) 𝑀 ) )