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Metamath Proof Explorer

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 )