Theorem estrcbas

Description: Set of objects of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020)

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

Proof

Step	Hyp	Ref	Expression
1		estrcbas.c	⊢ 𝐶 = ( ExtStrCat ‘ 𝑈 )
2		estrcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 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 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ }
6	3 4 5	strfv	⊢ ( 𝑈 ∈ 𝑉 → 𝑈 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
7	2 6	syl	⊢ ( 𝜑 → 𝑈 = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
8		eqidd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) )
9		eqidd	⊢ ( 𝜑 → ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) = ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) )
10	1 2 8 9	estrcval	⊢ ( 𝜑 → 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } )
11	10	fveq2d	⊢ ( 𝜑 → ( Base ‘ 𝐶 ) = ( Base ‘ { ⟨ ( Base ‘ ndx ) , 𝑈 ⟩ , ⟨ ( Hom ‘ ndx ) , ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( ( Base ‘ 𝑦 ) ↑m ( Base ‘ 𝑥 ) ) ) ⟩ , ⟨ ( comp ‘ ndx ) , ( 𝑣 ∈ ( 𝑈 × 𝑈 ) , 𝑧 ∈ 𝑈 ↦ ( 𝑔 ∈ ( ( Base ‘ 𝑧 ) ↑m ( Base ‘ ( 2nd ‘ 𝑣 ) ) ) , 𝑓 ∈ ( ( Base ‘ ( 2nd ‘ 𝑣 ) ) ↑m ( Base ‘ ( 1st ‘ 𝑣 ) ) ) ↦ ( 𝑔 ∘ 𝑓 ) ) ) ⟩ } ) )
12	7 11	eqtr4d	⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐶 ) )