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Metamath Proof Explorer

Theorem catchom

Description: Set of arrows of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017)

Ref Expression
Hypotheses catcbas.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
catcbas.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
catcbas.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
catchomfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
catchom.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
catchom.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion catchom ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 Func 𝑌 ) )

Proof

Step Hyp Ref Expression
1 catcbas.c ⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2 catcbas.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 catcbas.u ⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4 catchomfval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
5 catchom.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 catchom.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 1 2 3 4 catchomfval ⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) )
8 oveq12 ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 Func 𝑦 ) = ( 𝑋 Func 𝑌 ) )
9 8 adantl ⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑥 Func 𝑦 ) = ( 𝑋 Func 𝑌 ) )
10 ovexd ⊢ ( 𝜑 → ( 𝑋 Func 𝑌 ) ∈ V )
11 7 9 5 6 10 ovmpod ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 Func 𝑌 ) )