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Metamath Proof Explorer

Theorem relup

Description: The set of universal pairs is a relation. (Contributed by Zhi Wang, 25-Sep-2025)

Ref Expression
Assertion relup Rel ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
2 eqid ⊢ ( Base ‘ 𝐸 ) = ( Base ‘ 𝐸 )
3 eqid ⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
4 eqid ⊢ ( Hom ‘ 𝐸 ) = ( Hom ‘ 𝐸 )
5 eqid ⊢ ( comp ‘ 𝐸 ) = ( comp ‘ 𝐸 )
6 1 2 3 4 5 upfval ⊢ ( 𝐷 UP 𝐸 ) = ( 𝑓 ∈ ( 𝐷 Func 𝐸 ) , 𝑤 ∈ ( Base ‘ 𝐸 ) ↦ { ⟨ 𝑥 , 𝑚 ⟩ ∣ ( ( 𝑥 ∈ ( Base ‘ 𝐷 ) ∧ 𝑚 ∈ ( 𝑤 ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ) ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐷 ) ∀ 𝑔 ∈ ( 𝑤 ( Hom ‘ 𝐸 ) ( ( 1st ‘ 𝑓 ) ‘ 𝑦 ) ) ∃! 𝑘 ∈ ( 𝑥 ( Hom ‘ 𝐷 ) 𝑦 ) 𝑔 = ( ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ 𝑘 ) ( ⟨ 𝑤 , ( ( 1st ‘ 𝑓 ) ‘ 𝑥 ) ⟩ ( comp ‘ 𝐸 ) ( ( 1st ‘ 𝑓 ) ‘ 𝑦 ) ) 𝑚 ) ) } )
7 6 relmpoopab ⊢ Rel ( 𝐹 ( 𝐷 UP 𝐸 ) 𝑊 )