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Metamath Proof Explorer

Theorem comfval

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses comfffval.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
comfffval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
comfffval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
comfffval.x ⊢ · = ( comp ‘ 𝐶 )
comffval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
comffval.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
comffval.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
comfval.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
comfval.g ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
Assertion comfval ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) 𝐹 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )

Proof

Step Hyp Ref Expression
1 comfffval.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
2 comfffval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 comfffval.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 comfffval.x ⊢ · = ( comp ‘ 𝐶 )
5 comffval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 comffval.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 comffval.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8 comfval.f ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
9 comfval.g ⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
10 1 2 3 4 5 6 7 comffval ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) = ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) ) )
11 oveq12 ⊢ ( ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) → ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )
12 11 adantl ⊢ ( ( 𝜑 ∧ ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) ) → ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )
13 ovexd ⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) ∈ V )
14 10 12 9 8 13 ovmpod ⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) 𝐹 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )