Theorem isup

Description: The predicate "is a universal pair". (Contributed by Zhi Wang, 24-Sep-2025)

	Ref	Expression
Hypotheses	upfval.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
	upfval.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
	upfval.h	⊢ 𝐻 = ( Hom ‘ 𝐷 )
	upfval.j	⊢ 𝐽 = ( Hom ‘ 𝐸 )
	upfval.o	⊢ 𝑂 = ( comp ‘ 𝐸 )
	upfval2.w	⊢ ( 𝜑 → 𝑊 ∈ 𝐶 )
	upfval3.f	⊢ ( 𝜑 → 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 )
	isup.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	isup.m	⊢ ( 𝜑 → 𝑀 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑋 ) ) )
Assertion	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 ) ) )

Proof

Step	Hyp	Ref	Expression
1		upfval.b	⊢ 𝐵 = ( Base ‘ 𝐷 )
2		upfval.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
3		upfval.h	⊢ 𝐻 = ( Hom ‘ 𝐷 )
4		upfval.j	⊢ 𝐽 = ( Hom ‘ 𝐸 )
5		upfval.o	⊢ 𝑂 = ( comp ‘ 𝐸 )
6		upfval2.w	⊢ ( 𝜑 → 𝑊 ∈ 𝐶 )
7		upfval3.f	⊢ ( 𝜑 → 𝐹 ( 𝐷 Func 𝐸 ) 𝐺 )
8		isup.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
9		isup.m	⊢ ( 𝜑 → 𝑀 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑋 ) ) )
10	8 9	jca	⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ 𝑀 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑋 ) ) ) )
11	1 2 3 4 5 6 7	isuplem	⊢ ( 𝜑 → ( 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 ↔ ( ( 𝑋 ∈ 𝐵 ∧ 𝑀 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑋 ) ) ) ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑊 𝐽 ( 𝐹 ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑋 𝐻 𝑦 ) 𝑔 = ( ( ( 𝑋 𝐺 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑊 , ( 𝐹 ‘ 𝑋 ) ⟩ 𝑂 ( 𝐹 ‘ 𝑦 ) ) 𝑀 ) ) ) )
12	10 11	mpbirand	⊢ ( 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 ) ) )