This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem comffn

Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses comfffn.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
comfffn.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
comffn.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
comffn.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
comffn.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
comffn.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
Assertion comffn ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) Fn ( ( 𝑌 𝐻 𝑍 ) × ( 𝑋 𝐻 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 comfffn.o ⊢ 𝑂 = ( compf ‘ 𝐶 )
2 comfffn.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 comffn.h ⊢ 𝐻 = ( Hom ‘ 𝐶 )
4 comffn.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 comffn.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 comffn.z ⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
7 eqid ⊢ ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑍 ) 𝑓 ) ) = ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑍 ) 𝑓 ) )
8 ovex ⊢ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑍 ) 𝑓 ) ∈ V
9 7 8 fnmpoi ⊢ ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑍 ) 𝑓 ) ) Fn ( ( 𝑌 𝐻 𝑍 ) × ( 𝑋 𝐻 𝑌 ) )
10 eqid ⊢ ( comp ‘ 𝐶 ) = ( comp ‘ 𝐶 )
11 1 2 3 10 4 5 6 comffval ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) = ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑍 ) 𝑓 ) ) )
12 11 fneq1d ⊢ ( 𝜑 → ( ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) Fn ( ( 𝑌 𝐻 𝑍 ) × ( 𝑋 𝐻 𝑌 ) ) ↔ ( 𝑔 ∈ ( 𝑌 𝐻 𝑍 ) , 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ↦ ( 𝑔 ( ⟨ 𝑋 , 𝑌 ⟩ ( comp ‘ 𝐶 ) 𝑍 ) 𝑓 ) ) Fn ( ( 𝑌 𝐻 𝑍 ) × ( 𝑋 𝐻 𝑌 ) ) ) )
13 9 12 mpbiri ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) Fn ( ( 𝑌 𝐻 𝑍 ) × ( 𝑋 𝐻 𝑌 ) ) )