Theorem catcco

Description: Composition in the category of categories. (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 )
	catcco.o	⊢ · = ( comp ‘ 𝐶 )
	catcco.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	catcco.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	catcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	catcco.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 Func 𝑌 ) )
	catcco.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 Func 𝑍 ) )
Assertion	catcco	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) = ( 𝐺 ∘func 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		catcbas.c	⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2		catcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		catcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		catcco.o	⊢ · = ( comp ‘ 𝐶 )
5		catcco.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		catcco.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		catcco.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
8		catcco.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 Func 𝑌 ) )
9		catcco.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑌 Func 𝑍 ) )
10	1 2 3 4	catccofval	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) )
11		simprl	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ )
12	11	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 𝑣 ) = ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
13		op2ndg	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
14	5 6 13	syl2anc	⊢ ( 𝜑 → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
15	14	adantr	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ ⟨ 𝑋 , 𝑌 ⟩ ) = 𝑌 )
16	12 15	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 2nd ‘ 𝑣 ) = 𝑌 )
17		simprr	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → 𝑧 = 𝑍 )
18	16 17	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) = ( 𝑌 Func 𝑍 ) )
19	11	fveq2d	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Func ‘ 𝑣 ) = ( Func ‘ ⟨ 𝑋 , 𝑌 ⟩ ) )
20		df-ov	⊢ ( 𝑋 Func 𝑌 ) = ( Func ‘ ⟨ 𝑋 , 𝑌 ⟩ )
21	19 20	eqtr4di	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( Func ‘ 𝑣 ) = ( 𝑋 Func 𝑌 ) )
22		eqidd	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 𝑔 ∘func 𝑓 ) = ( 𝑔 ∘func 𝑓 ) )
23	18 21 22	mpoeq123dv	⊢ ( ( 𝜑 ∧ ( 𝑣 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝑧 = 𝑍 ) ) → ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) = ( 𝑔 ∈ ( 𝑌 Func 𝑍 ) , 𝑓 ∈ ( 𝑋 Func 𝑌 ) ↦ ( 𝑔 ∘func 𝑓 ) ) )
24	5 6	opelxpd	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
25		ovex	⊢ ( 𝑌 Func 𝑍 ) ∈ V
26		ovex	⊢ ( 𝑋 Func 𝑌 ) ∈ V
27	25 26	mpoex	⊢ ( 𝑔 ∈ ( 𝑌 Func 𝑍 ) , 𝑓 ∈ ( 𝑋 Func 𝑌 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ∈ V
28	27	a1i	⊢ ( 𝜑 → ( 𝑔 ∈ ( 𝑌 Func 𝑍 ) , 𝑓 ∈ ( 𝑋 Func 𝑌 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ∈ V )
29	10 23 24 7 28	ovmpod	⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) = ( 𝑔 ∈ ( 𝑌 Func 𝑍 ) , 𝑓 ∈ ( 𝑋 Func 𝑌 ) ↦ ( 𝑔 ∘func 𝑓 ) ) )
30		oveq12	⊢ ( ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) → ( 𝑔 ∘func 𝑓 ) = ( 𝐺 ∘func 𝐹 ) )
31	30	adantl	⊢ ( ( 𝜑 ∧ ( 𝑔 = 𝐺 ∧ 𝑓 = 𝐹 ) ) → ( 𝑔 ∘func 𝑓 ) = ( 𝐺 ∘func 𝐹 ) )
32		ovexd	⊢ ( 𝜑 → ( 𝐺 ∘func 𝐹 ) ∈ V )
33	29 31 9 8 32	ovmpod	⊢ ( 𝜑 → ( 𝐺 ( ⟨ 𝑋 , 𝑌 ⟩ · 𝑍 ) 𝐹 ) = ( 𝐺 ∘func 𝐹 ) )