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Metamath Proof Explorer

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 ) )