Theorem catbas

Description: The base of the category structure. (Contributed by Zhi Wang, 5-Nov-2025)

	Ref	Expression
Hypotheses	catbas.c	⊢ 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
	catbas.b	⊢ 𝐵 ∈ V
Assertion	catbas	⊢ 𝐵 = ( Base ‘ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		catbas.c	⊢ 𝐶 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
2		catbas.b	⊢ 𝐵 ∈ V
3		catstr	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ } Struct ⟨ 1 , ; 1 5 ⟩
4	1 3	eqbrtri	⊢ 𝐶 Struct ⟨ 1 , ; 1 5 ⟩
5		baseid	⊢ Base = Slot ( Base ‘ ndx )
6		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( Hom ‘ ndx ) , 𝐻 ⟩ , ⟨ ( comp ‘ ndx ) , · ⟩ }
7	6 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝐶
8	4 5 7	strfv	⊢ ( 𝐵 ∈ V → 𝐵 = ( Base ‘ 𝐶 ) )
9	2 8	ax-mp	⊢ 𝐵 = ( Base ‘ 𝐶 )