Theorem uptpos

Description: Rewrite the predicate of universal property in the form of opposite functor. (Contributed by Zhi Wang, 4-Nov-2025)

	Ref	Expression
Hypotheses	oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
	uptpos.h	⊢ ( 𝜑 → tpos 𝐺 = 𝐻 )
Assertion	uptpos	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , tpos 𝐻 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
2		uptpos.h	⊢ ( 𝜑 → tpos 𝐺 = 𝐻 )
3	1 2	uptposlem	⊢ ( 𝜑 → tpos 𝐻 = 𝐺 )
4	3	opeq2d	⊢ ( 𝜑 → ⟨ 𝐹 , tpos 𝐻 ⟩ = ⟨ 𝐹 , 𝐺 ⟩ )
5	4	oveq1d	⊢ ( 𝜑 → ( ⟨ 𝐹 , tpos 𝐻 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) = ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) )
6	5	breqd	⊢ ( 𝜑 → ( 𝑋 ( ⟨ 𝐹 , tpos 𝐻 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 ↔ 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 ) )
7	1 6	mpbird	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , tpos 𝐻 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )