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 ) )