Theorem up1st2nd

Description: Rewrite the universal property predicate with separated parts. (Contributed by Zhi Wang, 23-Oct-2025)

	Ref	Expression
Hypothesis	up1st2nd.1	⊢ ( 𝜑 → 𝑋 ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
Assertion	up1st2nd	⊢ ( 𝜑 → 𝑋 ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		up1st2nd.1	⊢ ( 𝜑 → 𝑋 ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
2		relfunc	⊢ Rel ( 𝐷 Func 𝐸 )
3		df-br	⊢ ( 𝑋 ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 ↔ ⟨ 𝑋 , 𝑀 ⟩ ∈ ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) )
4	1 3	sylib	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑀 ⟩ ∈ ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) )
5		eqid	⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 )
6	5	uprcl	⊢ ( ⟨ 𝑋 , 𝑀 ⟩ ∈ ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) → ( 𝐹 ∈ ( 𝐷 Func 𝐸 ) ∧ 𝑊 ∈ ( Base ‘ 𝐸 ) ) )
7	4 6	syl	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐷 Func 𝐸 ) ∧ 𝑊 ∈ ( Base ‘ 𝐸 ) ) )
8	7	simpld	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐷 Func 𝐸 ) )
9		1st2nd	⊢ ( ( Rel ( 𝐷 Func 𝐸 ) ∧ 𝐹 ∈ ( 𝐷 Func 𝐸 ) ) → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
10	2 8 9	sylancr	⊢ ( 𝜑 → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
11	10	oveq1d	⊢ ( 𝜑 → ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) = ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) )
12	11 1	breqdi	⊢ ( 𝜑 → 𝑋 ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )