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Metamath Proof Explorer

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 ) ) )