Theorem catcbas

Description: Set of objects of 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 )
Assertion	catcbas	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Cat ) )

Proof

Step	Hyp	Ref	Expression
1		catcbas.c	⊢ 𝐶 = ( CatCat ‘ 𝑈 )
2		catcbas.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
3		catcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
4		eqidd	⊢ ( 𝜑 → ( 𝑈 ∩ Cat ) = ( 𝑈 ∩ Cat ) )
5		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑈 ∩ Cat ) , 𝑦 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑥 Func 𝑦 ) ) = ( 𝑥 ∈ ( 𝑈 ∩ Cat ) , 𝑦 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑥 Func 𝑦 ) ) )
6		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( ( 𝑈 ∩ Cat ) × ( 𝑈 ∩ Cat ) ) , 𝑧 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) = ( 𝑣 ∈ ( ( 𝑈 ∩ Cat ) × ( 𝑈 ∩ Cat ) ) , 𝑧 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) )
7	1 3 4 5 6	catcval	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Cat ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Cat ) , 𝑦 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑥 Func 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Cat ) × ( 𝑈 ∩ Cat ) ) , 𝑧 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } )
8		catstr	⊢ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Cat ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Cat ) , 𝑦 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑥 Func 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Cat ) × ( 𝑈 ∩ Cat ) ) , 𝑧 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
9		baseid	⊢ Base = Slot ( Base ‘ ndx )
10		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Cat ) ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , ( 𝑈 ∩ Cat ) ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ ( 𝑈 ∩ Cat ) , 𝑦 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑥 Func 𝑦 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( ( 𝑈 ∩ Cat ) × ( 𝑈 ∩ Cat ) ) , 𝑧 ∈ ( 𝑈 ∩ Cat ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑣 ) Func 𝑧 ) , 𝑓 ∈ ( Func ‘ 𝑣 ) ↦ ( 𝑔 ∘func 𝑓 ) ) ) ⟩ }
11		inex1g	⊢ ( 𝑈 ∈ 𝑉 → ( 𝑈 ∩ Cat ) ∈ V )
12	3 11	syl	⊢ ( 𝜑 → ( 𝑈 ∩ Cat ) ∈ V )
13	7 8 9 10 12 2	strfv3	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Cat ) )