Theorem catccofval

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 )
Assertion	catccofval	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 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	1 2 3	catcbas	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Cat ) )
6		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
7	1 2 3 6	catchomfval	⊢ ( 𝜑 → ( Hom ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 Func 𝑦 ) ) )
8		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) )
9	1 3 5 7 8	catcval	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } )
10	9	fveq2d	⊢ ( 𝜑 → ( comp ‘ 𝐶 ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } ) )
11	2	fvexi	⊢ 𝐵 ∈ V
12	11 11	xpex	⊢ ( 𝐵 × 𝐵 ) ∈ V
13	12 11	mpoex	⊢ ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ∈ V
14		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
15		ccoid	⊢ comp = Slot ( comp ‘ ndx )
16		snsstp3	⊢ { ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ }
17	14 15 16	strfv	⊢ ( ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ∈ V → ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } ) )
18	13 17	ax-mp	⊢ ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) = ( comp ‘ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , ( Hom ‘ 𝐶 ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } )
19	10 4 18	3eqtr4g	⊢ ( 𝜑 → · = ( 𝑣 ∈ ( 𝐵 × 𝐵 ) , 𝑧 ∈ 𝐵 ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) )