Theorem comfval2

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 )
	comfval2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
	comfval2.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
Assertion	comfval2	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) 𝐹 ) = ( 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		comfval2.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) )
9		comfval2.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 𝐻 𝑍 ) )
10		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
11	3 2 10 5 6	homfval	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
12	8 11	eleqtrd	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 ( Hom ‘ 𝐶 ) 𝑌 ) )
13	3 2 10 6 7	homfval	⊢ ( 𝜑 → ( 𝑌 𝐻 𝑍 ) = ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) )
14	9 13	eleqtrd	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 ( Hom ‘ 𝐶 ) 𝑍 ) )
15	1 2 10 4 5 6 7 12 14	comfval	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ 𝑂 𝑍 ) 𝐹 ) = ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) )