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Metamath Proof Explorer

Theorem homarcl2

Description: Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses homahom.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
homarcl2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
Assertion homarcl2 ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 homahom.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
2 homarcl2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 elfvdm ⊢ ( 𝐹 ∈ ( 𝐻 ‘ ⟨ 𝑋 , 𝑌 ⟩ ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom 𝐻 )
4 df-ov ⊢ ( 𝑋 𝐻 𝑌 ) = ( 𝐻 ‘ ⟨ 𝑋 , 𝑌 ⟩ )
5 3 4 eleq2s ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom 𝐻 )
6 1 homarcl ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐶 ∈ Cat )
7 1 2 6 homaf ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → 𝐻 : ( 𝐵 × 𝐵 ) ⟶ 𝒫 ( ( 𝐵 × 𝐵 ) × V ) )
8 7 fdmd ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → dom 𝐻 = ( 𝐵 × 𝐵 ) )
9 5 8 eleqtrd ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
10 opelxp ⊢ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )
11 9 10 sylib ⊢ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) → ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) )