Theorem uprcl3

Description: Reverse closure for the class of universal property. (Contributed by Zhi Wang, 25-Sep-2025)

	Ref	Expression
Hypotheses	uprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
	uprcl3.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
Assertion	uprcl3	⊢ ( 𝜑 → 𝑊 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		uprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
2		uprcl3.c	⊢ 𝐶 = ( Base ‘ 𝐸 )
3		df-br	⊢ ( 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 ↔ ⟨ 𝑋 , 𝑀 ⟩ ∈ ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) )
4	3	biimpi	⊢ ( 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 → ⟨ 𝑋 , 𝑀 ⟩ ∈ ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) )
5	2	uprcl	⊢ ( ⟨ 𝑋 , 𝑀 ⟩ ∈ ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) → ( ⟨ 𝐹 , 𝐺 ⟩ ∈ ( 𝐷 Func 𝐸 ) ∧ 𝑊 ∈ 𝐶 ) )
6	5	simprd	⊢ ( ⟨ 𝑋 , 𝑀 ⟩ ∈ ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) → 𝑊 ∈ 𝐶 )
7	1 4 6	3syl	⊢ ( 𝜑 → 𝑊 ∈ 𝐶 )