Theorem up1st2ndr

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

	Ref	Expression
Hypotheses	up1st2ndr.1	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐷 Func 𝐸 ) )
	up1st2ndr.2	⊢ ( 𝜑 → 𝑋 ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
Assertion	up1st2ndr	⊢ ( 𝜑 → 𝑋 ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )

Proof

Step	Hyp	Ref	Expression
1		up1st2ndr.1	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐷 Func 𝐸 ) )
2		up1st2ndr.2	⊢ ( 𝜑 → 𝑋 ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )
3		relfunc	⊢ Rel ( 𝐷 Func 𝐸 )
4		1st2nd	⊢ ( ( Rel ( 𝐷 Func 𝐸 ) ∧ 𝐹 ∈ ( 𝐷 Func 𝐸 ) ) → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
5	3 1 4	sylancr	⊢ ( 𝜑 → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
6	5	oveq1d	⊢ ( 𝜑 → ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) = ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) )
7	6	eqcomd	⊢ ( 𝜑 → ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ ( 𝐷 UP 𝐸 ) 𝑊 ) = ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) )
8	7 2	breqdi	⊢ ( 𝜑 → 𝑋 ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 ) 𝑀 )