Metamath Proof Explorer

Theorem setcbas

Description: Set of objects of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017)

		Ref	Expression
	Hypotheses	setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	Assertion	setcbas	⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
2		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
4		baseid	⊢ Base = Slot ( Base ‘ ndx )
5		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ }
6	3 4 5	strfv	⊢ ( 𝑈 ∈ 𝑉 → 𝑈 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
7	2 6	syl	⊢ ( 𝜑 → 𝑈 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
8		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) )
9		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )
10	1 2 8 9	setcval	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
11	10	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑_m 𝑥 ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( 𝑧 ↑_m ( 2^nd ‘ 𝑣 ) ) , 𝑓 ∈ ( ( 2^nd ‘ 𝑣 ) ↑_m ( 1^st ‘ 𝑣 ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
12	7 11	eqtr4d	⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) )