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Metamath Proof Explorer

Theorem homahom2

Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses homahom.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
homahom.j ⊢ 𝐽 = ( Hom ‘ 𝐶 )
Assertion homahom2 ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 → 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) )

Proof

Step Hyp Ref Expression
1 homahom.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
2 homahom.j ⊢ 𝐽 = ( Hom ‘ 𝐶 )
3 df-br ⊢ ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ ⟨ 𝑍 , 𝐹 ⟩ ∈ ( 𝑋 𝐻 𝑌 ) )
4 eqid ⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
5 1 homarcl ⊢ ( ⟨ 𝑍 , 𝐹 ⟩ ∈ ( 𝑋 𝐻 𝑌 ) → 𝐶 ∈ Cat )
6 1 4 homarcl2 ⊢ ( ⟨ 𝑍 , 𝐹 ⟩ ∈ ( 𝑋 𝐻 𝑌 ) → ( 𝑋 ∈ ( Base ‘ 𝐶 ) ∧ 𝑌 ∈ ( Base ‘ 𝐶 ) ) )
7 6 simpld ⊢ ( ⟨ 𝑍 , 𝐹 ⟩ ∈ ( 𝑋 𝐻 𝑌 ) → 𝑋 ∈ ( Base ‘ 𝐶 ) )
8 6 simprd ⊢ ( ⟨ 𝑍 , 𝐹 ⟩ ∈ ( 𝑋 𝐻 𝑌 ) → 𝑌 ∈ ( Base ‘ 𝐶 ) )
9 1 4 5 2 7 8 elhoma ⊢ ( ⟨ 𝑍 , 𝐹 ⟩ ∈ ( 𝑋 𝐻 𝑌 ) → ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) ) )
10 3 9 sylbi ⊢ ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 → ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) ) )
11 10 ibi ⊢ ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 → ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) )
12 11 simprd ⊢ ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 → 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) )