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Metamath Proof Explorer

Theorem comffval2

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

Ref Expression
Hypotheses comfffval2.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
comfffval2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
comfffval2.h ⊢ 𝐻 = ( Homf ‘ 𝐶 )
comfffval2.x ⊢ · = ( comp ‘ 𝐶 )
comffval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
comffval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
comffval2.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion comffval2 ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) = ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) ) )

Proof

Step Hyp Ref Expression
1 comfffval2.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
2 comfffval2.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 comfffval2.h ⊢ 𝐻 = ( Homf ‘ 𝐶 )
4 comfffval2.x ⊢ · = ( comp ‘ 𝐶 )
5 comffval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 comffval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 comffval2.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8 eqid ⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
9 1 2 8 4 5 6 7 comffval ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) = ( 𝑔 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) , 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) ) )
10 3 2 8 6 7 homfval ⊢ ( 𝜑 → ( 𝑌 𝐻 𝑍 ) = ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) )
11 3 2 8 5 6 homfval ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
12 eqidd ⊢ ( 𝜑 → ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) = ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) )
13 10 11 12 mpoeq123dv ⊢ ( 𝜑 → ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) ) = ( 𝑔 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) , 𝑓 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) ) )
14 9 13 eqtr4d ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) = ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝑓 ) ) )